On the convexity of Berezin range and Berezin radius inequalities via a class of seminorms
Augustine A., Hiran Das P., Bhunia P., Shankar P.
Article, Bulletin des Sciences Mathematiques, 2026, DOI Link
View abstract ⏷
This paper introduces a new family of seminorms, say σμ-Berezin norm on the space of all bounded linear operators B(H) defined on a reproducing kernel Hilbert space H on a nonempty set Ω, namely, for each μ∈[0,1] and p≥1, [Formula presented] where T∈B(H) and σμ is an interpolation path of the symmetric mean σ. We investigate many fundamental properties of the σμ-Berezin norm and develop several inequalities associated with it. Utilizing these inequalities, we derive improved bounds for the Berezin radius of bounded linear operators, enhancing previously known estimates. Furthermore, we study the convexity of the Berezin range of a class of composition operators and weighted shift operators on both the Hardy space and the Bergman space.
Strengthening of spectral radius, numerical radius, and Berezin radius inequalities
Article, Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas, 2026, DOI Link
View abstract ⏷
We present a numerical radius bound for n×n operator matrices that improves the bound of Abu-Omar and Kittaneh (Linear Algebra Appl 468:18–26, 2015). As a significant application, we derive an estimate for the numerical radius of the Kronecker products A⊗B, where A is an n×n matrix and B is a bounded linear operator. This result refines Holbrook’s classical bound w(A⊗B)≤w(A)‖B‖ in the special case when all entries of A are non-negative. In addition, we establish spectral radius inequalities for the sums, products, and commutators of operators, improving upon the bounds of Kittaneh (Proc Am Math Soc 134:385–390, 2006) and Abu-Omar and Kittaneh (Stud Math 216(1):69–75, 2013). We further obtain an estimate for the zeros of an algebraic equation via Frobenius companion matrix, strengthening the bound of Abdurakhmanov (Mat Sb (N.S.) 131(173)(1):40–51, 126, 1986; translation in Math. USSR-Sb. 59(1):39–51, 1988). Furthermore, the Berezin radius inequalities are established, supported by several illustrative examples.
Buzano type inequalities in semi-Hilbertian spaces with applications
Article, Annali dell'Universita di Ferrara, 2026, DOI Link
View abstract ⏷
By developing Buzano type inequalities in semi-Hilbertian spaces, we obtain several A0-numerical radius inequalities for 2×2 block matrices, where A0=A00A is a 2×2 diagonal block matrix, whose each diagonal entry is a positive bounded linear operator A on a complex Hilbert space. These inequalities improve and generalize some previously related inequalities. We also deduce several improved A-numerical radius inequalities for semi-Hilbertian space operators.
Perspectives on the ρ-operator radius
Bhunia P., Moslehian M.S., Zamani A.
Article, Journal of Mathematical Analysis and Applications, 2026, DOI Link
View abstract ⏷
Let ρ∈(0,2] and let wρ(X) be the ρ-operator radius of a Hilbert space operator X. Using techniques involving the Kronecker product, it is shown that [Formula presented] where w(X) is the numerical radius of X. These bounds for wρ(X) are sharper than those presented by J. A. R. Holbrook. Furthermore, the cases of equality are investigated. We prove that the ρ-operator radius exposes certain operators as projections. We establish new inequalities for the ρ-operator radius, focusing on the sum and product of operators. For the generalized Aluthge transform X˜t of an operator X, we prove the inequality: [Formula presented] The derived inequalities extend and generalize several well-known results for the classical operator norm and numerical radius.
Numerical radius and ℓp operator norm of Kronecker products and Schur powers: inequalities and equalities
Bhunia P., Sakharam Damase S., Khare A.
Article, Linear Algebra and Its Applications, 2026, DOI Link
View abstract ⏷
Suppose A=[aij]∈Mn(C) is a complex n×n matrix and B∈B(H) is a bounded linear operator on a complex Hilbert space H. We show that w(A⊗B)≤w(C), where w(⋅) denotes the numerical radius and C=[cij] with cij=w([0aijaji0]⊗B). This refines Holbrook's classical bound w(A⊗B)≤w(A)‖B‖ (1969) [31], when all entries of A are non-negative. If moreover aii≠0 ∀i, we prove that w(A⊗B)=w(A)‖B‖ if and only if w(B)=‖B‖. We then extend these and other results to the more general setting of semi-Hilbertian spaces induced by a positive operator. In the reverse direction, we also specialize these results to Kronecker products and hence to Schur/entrywise products, of matrices: (1)(a) We first provide an alternate proof (using w(A)) of a result of Goldberg and Zwas (1974) [24] that if the spectral norm of A equals its spectral radius, then each Jordan block for each maximum-modulus eigenvalue must be 1×1 (“partial diagonalizability”). (b) Using our approach, we further show given m≥1 that w(A∘m)≤wm(A) – we also characterize when equality holds here. (2) We provide upper and lower bounds for the ℓp operator norm and the numerical radius of A⊗B for all A∈Mn(C), which become equal when restricted to doubly stochastic matrices A. Finally, using these bounds we obtain an improved estimation for the roots of an arbitrary complex polynomial.
Euclidean operator radius and numerical radius bounds via the notion of angle between two vectors
Bhunia P., Jana S., Kittaneh F.
Article, Journal of Analysis, 2025, DOI Link
View abstract ⏷
This paper aims to establish new upper and lower bounds for the Euclidean operator radius of a pair of bounded linear operators on a complex Hilbert space and derive novel upper bounds for the numerical radius, by utilizing the notion of angle between two vectors. Among other numerical radius bounds, it is shown that (Formula presented.) where w(T) and denote the numerical radius and the operator norm of a bounded linear operator T, respectively.
Numerical radius inequalities of operator matrices
Article, Indian Journal of Pure and Applied Mathematics, 2025, DOI Link
View abstract ⏷
Suppose [Aij] is an n×n operator matrix, where each Aij is a bounded linear operator on a complex Hilbert space H. Among other inequalities, it is shown that w([Aij])≤w([aij]), where [aij] is an n×n matrix with (Formula presented.) This numerical radius bound refines a well known bound by Abu-Omar and Kittaneh [Linear Algebra Appl. 468 (2015), 18–26]. We use these estimates to derive several numerical radius inequalities and equalities for 2×2 operator matrices. Applying these inequalities, we also deduce several numerical radius bounds for a bounded linear operator, the product of two operators and the commutator of operators. In particular, it is shown that (Formula presented.) where A is a bounded linear operator on H. This bound refines as well as generalizes the well known bounds.
Schatten p-Norm and Numerical Radius Inequalities with Applications
Article, Results in Mathematics, 2025, DOI Link
View abstract ⏷
We develop a new refinement of the Kato’s inequality and using this refinement we obtain several upper bounds for the numerical radius of a bounded linear operator as well as the product of operators, which improve the well known existing bounds. Further, we obtain a necessary and sufficient condition for the positivity of 2×2 certain block matrices and using this condition we deduce an upper bound for the numerical radius involving a contraction operator. Furthermore, we study the Schatten p-norm inequalities for the sum of two n×n complex matrices via singular values, and from the inequalities we obtain the p-numerical radius and the classical numerical radius bounds. We show that for every p>0, the p-numerical radius wp(·):Mn(C)→R satisfies wp(T)≤12|T|2(1-t)+|T∗|2(1-t)‖|T|2t+|T∗|2t‖p/2 for all t∈[0,1]. Considering p→∞, we get a nice refinement of the well known classical numerical radius bound w(T)≤12T∗T+TT∗. As an application of the Schatten p-norm inequalities we develop a bound for the energy of a graph. We show that E(G)≥2mmax1≤i≤n∑j,vi∼vjdj, where E(G) is the energy of a simple graph G with m edges and n vertices v1,v2,…,vn such that degree of vi is di for each i=1,2,…,n.
REFINED INEQUALITIES FOR THE NUMERICAL RADIUS OF HILBERT SPACE OPERATORS
Article, Rocky Mountain Journal of Mathematics, 2025, DOI Link
View abstract ⏷
We present some new upper and lower bounds for the numerical radius of bounded linear operators on a complex Hilbert space and show that the bounds are stronger than the existing ones. In particular, we prove that if A is a bounded linear operator on a complex Hilbert space H and if ℜ(A), ℑ(A) are the real part, the imaginary part of A, respectively, then (Formula presented) and (Formula presented) where w(· ) and ∥ · ∥ denote the numerical radius and the operator norm, respectively. Further, we obtain refinements of the inequalities for the numerical radius of the product of two operators. Finally, as an application of the second inequality mentioned above, we obtain an improvement of upper bound for the numerical radius of the commutators of operators.
Improved numerical radius bounds using the Moore-Penrose inverse
Bhunia P., Kittaneh F., Sahoo S.
Article, Linear Algebra and Its Applications, 2025, DOI Link
View abstract ⏷
Using the Moore-Penrose inverse of a bounded linear operator, we obtain few bounds for the numerical radius, which improve the classical ones. Applying these improvements, we study equality conditions of the existing bounds. It is shown that if T is a bounded linear operator with closed range, then [Formula presented] For a finite-dimensional space operator T, this improvement is proper if and only if Range(T)∩Range(T⁎)={0}. Clearly, if ‖TT†+T†T‖=1, then [Formula presented]. Among other results, we obtain inner product inequalities for the sum of operators, and as an application of these inequalities, we deduce relevant operator norm and numerical radius bounds.
Norm inequalities for Hilbert space operators with applications
Article, Linear Algebra and Its Applications, 2025, DOI Link
View abstract ⏷
Several unitarily invariant norm inequalities and numerical radius inequalities for Hilbert space operators are studied. We investigate some necessary and sufficient conditions for the parallelism of two bounded operators. For a finite rank operator A, it is shown that ‖A‖p≤(rankA)1/2p‖A‖2p≤(rankA)(2p−1)/2p2‖A‖2p2,for all p≥1 where ‖⋅‖p is the Schatten p-norm. If {λn(A)} is a listing of all non-zero eigenvalues (with multiplicity) of a compact operator A, then we show that [Formula presented] which improves the classical Weyl's inequality ∑n|λn(A)|p≤‖A‖pp [Proc. Nat. Acad. Sci. USA 1949]. For an n×n matrix A, we show that the function p→n−1/p‖A‖p is monotone increasing on p≥1, complementing the well known decreasing nature of p→‖A‖p. As an application of these inequalities, we provide an upper bound for the sum of the absolute values of the zeros of a complex polynomial. As another application we provide a refined upper bound for the energy of a graph G, namely, E(G)≤2m(rank Adj(G)), where m is the number of edges, improving on a bound by McClelland in 1971.
Inequalities for linear functionals and numerical radii on C∗-algebras
Article, Acta Mathematica Hungarica, 2025, DOI Link
View abstract ⏷
Let A be a unital C∗-algebra with unit e.We develop several inequalities for a positive linear functional f on A and obtain several bounds for the numerical radius v(a) of an element a∈A.Among other inequalities, we show that if ak,bk,xk∈A, r∈N and f(e)=1, then (Formula presented.) We find several equivalent conditions for v(a)=‖a‖2 and v2(a)=14‖a∗a+aa∗‖.We prove that v2(a)=14‖a∗a+aa∗‖ (resp., v(a)=‖a‖2) if and only if (Formula presented.) (resp., S12‖a‖⊆V(a)⊆D12‖a‖),where V(a) is the numerical range of a and Dk (resp., Sk) denotes the circular disk (resp., semi-circular disk) with center at the origin and radius k. We also study inequalities for the (α,β)-normal elements in A.
Numerical radius and spectral radius inequalities with an estimation for roots of a polynomial
Article, Indian Journal of Pure and Applied Mathematics, 2025, DOI Link
View abstract ⏷
Suppose A is a bounded linear operator defined on a complex Hilbert space. Among other numerical radius inequalities, it is proved (by using the Aluthge transform A~ of A) that (Formula presented.) where w(A) is the numerical radius of A. This numerical radius bound improves the well known existing bound (Formula presented.) Additionally, we explore the spectral radius bounds of the sum, product and commutator of bounded linear operators. Furthermore, by using the spectral radius bound for the sum of two operators, we provide an estimation for the roots of a complex polynomial.
A New Norm on the Space of Reproducing Kernel Hilbert Space Operators and Berezin Number Inequalities
Article, Complex Analysis and Operator Theory, 2025, DOI Link
View abstract ⏷
In this note, we introduce a novel norm, termed the t-Berezin norm, on the algebra of all bounded linear operators defined on a reproducing kernel Hilbert space H(Ω) as (Formula presented.) where A∈B(H(Ω)) is a bounded linear operator. This norm characterizes those invertible operators which are also unitary. Using this newly defined norm, we establish various upper bounds for the Berezin number, thereby refining the existing results. Additionally, we derive several sharp bounds for the Berezin number of an operator via the Orlicz function.
Improved bounds for the numerical radius via a new norm on B(H)
Article, Georgian Mathematical Journal, 2025, DOI Link
View abstract ⏷
In this paper, we introduce a new norm, christened the t-operator norm, on the space of all bounded linear operators defined on a complex Hilbert space H as (Formula Prasented), where x, y ϵ H and t ϵ [ 0, 1 ]. This norm satisfies 1/2 Tt ≤ w (T) ≤ Tt 12 and we explore its properties. This norm characterizes those invertible operators that are also unitary. We obtain various inequalities involving the t-operator norm and the usual operator norm. We show that w (T) ≤ min t ϵ improves the existing bounds w (T) ≤ 1/2 w(T) (see [F. Kittaneh, A numerical radius inequality and an estimate for the numerical radius of the Frobenius companion matrix, Studia Math. 158 2003, 1, 11-17]) and w (T) ≤ 1/2 T ∗ (see [F. Kittaneh, Numerical radius inequalities for Hilbert space operators, Studia Math. 168 2005, 1, 73-80]). We show that (T) ≤ min | λ | = 1 . Further, we study the t-operator norm of operator matrices.
An improvement of Schrödinger’s uncertainty relation
Article, Physics Letters, Section A: General, Atomic and Solid State Physics, 2025, DOI Link
View abstract ⏷
Quantum uncertainty relations are mathematical inequalities that provide lower bounds on the products of the standard deviations of observables-represented by bounded or unbounded self-adjoint linear operators. In this note, we present an enhanced version of Schrödinger's uncertainty relation by incorporating the angle between two vectors. Furthermore, we derive multi-observable uncertainty relations, improving upon existing results. In addition, we investigate a sum uncertainty relation, extending the scope of our analysis.
A notion of the Cartesian decomposition and P-numerical radius bounds
Barik S., Bhunia P., Paul K.
Article, Journal of Analysis, 2025, DOI Link
View abstract ⏷
In this note, we introduce the notion of P-generalized Cartesian decomposition of operators in a semi-Hilbertian space induced by a positive operator P acting on a Hilbert space. Using this we obtain several generalizations of known P-numerical radius inequalities, which improve on the existing ones. Furthermore, we discuss characterizations for the equality of existing P-numerical radius inequalities.
A generalized mixed Schwarz inequality and its application to the numerical radius
Ren Y., Ighachane M.A., Bhunia P.
Article, Journal of Pseudo-Differential Operators and Applications, 2025, DOI Link
View abstract ⏷
In this paper, by employing the positivity of certain block operator matrices, we establish a new class of mixed Schwarz-type inequalities. Our results provide a unified framework that not only extends but also refines several classical inequalities in operator theory. In particular, the obtained inequalities encompass and generalize well-known results such as the mixed Schwarz inequality of Kato, the functional inequality of Kittaneh, and Furuta’s extension involving mixed operator powers. As an application, we prove several new numerical radius inequalities, which provide improved estimates and unify existing results in this direction. These contributions highlight the versatility of block operator techniques in deriving operator inequalities that unify and extend a wide range of known results in the literature.
Numerical radius inequalities of bounded linear operators and (α,β)-normal operators
Article, Acta Scientiarum Mathematicarum, 2025, DOI Link
View abstract ⏷
We obtain various upper bounds for the numerical radius w(T) of a bounded linear operator T defined on a complex Hilbert space H, by developing the upper bounds for the α-norm of T, which is defined as ‖T‖α=supα|⟨Tx,x⟩|2+(1-α)‖Tx‖2:x∈H,‖x‖=1 for 0≤α≤1. Further, we prove that (Formula presented.) For 0≤α≤1≤β, the operator T is called (α,β)-normal if α2T∗T≤TT∗≤β2T∗T holds. Note that every invertible operator is an (α,β)-normal operator for suitable values of α and β. Among other lower bounds for the numerical radius of an (α,β)-normal operator T, we show that (Formula presented.) where ℜ(T) and ℑ(T) are the real part and imaginary part of T, respectively.
Berezin number and Berezin norm inequalities for operator matrices
Bhunia P., Sen A., Barik S., Paul K.
Article, Linear and Multilinear Algebra, 2024, DOI Link
View abstract ⏷
We establish new upper bounds for the Berezin number and Berezin norm of operator matrices, which are refinements of existing bounds. Among other bounds, we prove that if (Formula presented.) is an (Formula presented.) operator matrix with (Formula presented.) for (Formula presented.), then (Formula presented.) and (Formula presented.) where (Formula presented.) (Formula presented.) if i<j and (Formula presented.) if i>j. We also provide examples which illustrate these bounds for some concrete operators acting on the Hardy-Hilbert space.
Refinements of generalized Euclidean operator radius inequalities of 2-tuple operators
Article, Filomat, 2024, DOI Link
View abstract ⏷
We develop several upper and lower bounds for the A-Euclidean operator radius of 2-tuple operators admitting A-adjoint, and show that they refine the earlier related bounds. As an application of the bounds developed here, we obtain sharper A-numerical radius bounds.
A-Numerical Radius of Semi-Hilbert Space Operators
Guesba M., Bhunia P., Paul K.
Article, Journal of Convex Analysis, 2024,
View abstract ⏷
Let A =(A00A) be a 2 × 2 diagonal operator matrix whose each diagonal entry is a positive bounded linear operator A acting on a complex Hilbert space H. Let T, S and R be bounded linear operators on H admitting A-adjoints, where T and R are A-positive. By considering an A-positive 2×2 operator matrix (TSS#AR), we develop several upper bounds for the A-numerical radius of S. Applying these upper bounds we obtain new A-numerical radius bounds for the product and the sum of arbitrary operators which admit A-adjoints. Related other inequalities are also derived.
Numerical radius bounds for certain operators
Article, Indian Journal of Pure and Applied Mathematics, 2024, DOI Link
View abstract ⏷
We provide sharp bounds for the numerical radius of bounded linear operators defined on a complex Hilbert space. We also provide sharp bounds for the numerical radius of AαXB1-α, AαXBα and the Heinz means of operators, where A, B, X are bounded linear operators with A,B≥0 and 0≤α≤1. Further, we study the A-numerical radius inequalities for semi-Hilbertian space operators. We prove that wA(T)≤1-12n-11/n‖T‖A when ATn=0 for some least positive integer n. Some equalities for the A-numerical radius inequalities are also studied.
Improved bounds for the numerical radius via polar decomposition of operators
Article, Linear Algebra and Its Applications, 2024, DOI Link
View abstract ⏷
Using the polar decomposition of a bounded linear operator A defined on a complex Hilbert space, we obtain several numerical radius inequalities of the operator A, which generalize and improve the earlier related ones. Among other bounds, we show that if w(A) is the numerical radius of A, then [Formula presented] for all t∈[0,1]. Also, we obtain some upper bounds for the numerical radius involving the spectral radius and the Aluthge transform of operators. It is shown that [Formula presented] where A˜=|A|1/2U|A|1/2 is the Aluthge transform of A and A=U|A| is the polar decomposition of A. Other related results are also provided.
Berezin number inequalities via positivity of 2×2 block matrices
Guesba M., Bhunia P., Paul K.
Article, Operators and Matrices, 2024, DOI Link
View abstract ⏷
Suppose B(H (Ω)) is the set of all bounded linear operators acting on a reproducing kernel Hilbert space H (Ω). Applying the positivity criteria of 2×2 block matrices, we develop several new upper bounds for the Berezin number of operators in B(H (Ω)) involving Berezin norm, which are better than the earlier ones. Among other results, we obtain that if T,S ∈ B(H (Ω)) and 0 <α< 1, then
Generalized Cartesian decomposition and numerical radius inequalities
Article, Rendiconti del Circolo Matematico di Palermo, 2024, DOI Link
View abstract ⏷
Let T={λ∈C:∣λ∣=1}. Every linear operator T on a complex Hilbert space H can be decomposed as (Formula presented.) designated as the generalized Cartesian decomposition of T. Using the generalized Cartesian decomposition we obtain several lower and upper bounds for the numerical radius of bounded linear operators which refine the existing bounds. We prove that if T is a bounded linear operator on H, then (Formula presented.) This improves the existing bounds w(T)≥12‖T‖, w(T)≥‖Re(T)‖, w(T)≥‖Im(T)‖ and so w2(T)≥14‖T∗T+TT∗‖, where Re(T) and Im(T) denote the the real part and the imaginary part of T, respectively. Further, using a lower bound for the numerical radius of a bounded linear operator, we develop upper bounds for the numerical radius of the commutator of operators which generalize and improve on the existing ones.
Norm inequalities in L(X) and a geometric constant
Article, Banach Journal of Mathematical Analysis, 2024, DOI Link
View abstract ⏷
We introduce a new norm (say α-norm) on L(X), the space of all bounded linear operators defined on a normed linear space X. We explore various properties of the α-norm. In addition, we study several equalities and inequalities of the α-norm of operators on X. As an application, we obtain an upper bound for the numerical radius of product of operators, which improves a well-known upper bound of the numerical radius for sectorial matrices. We present the α-norm of operators by using the extreme points of the unit ball of the corresponding spaces. Furthermore, we define a geometric constant (say α-index) associated with X and study properties of the α-index. In particular, we obtain the exact value of the α-index for some polyhedral spaces and complex Hilbert space. Finally, we study the α-index of ℓp-sum of normed linear spaces.
On the convergence of some spectral characteristics of the converging operator sequences
Bhunia P., Ipek Al P., Ismailov Z.I.
Article, Proceedings of the Indian Academy of Sciences: Mathematical Sciences, 2024, DOI Link
View abstract ⏷
Convergence of the differences between operator norm and spectral radius, operator norm and numerical radius, numerical radius and spectral radius, operator norm and Crawford number, operator norm and subspectral radius of complex Hilbert space operator sequences (which are uniformly convergent) has been investigated. Also, an inequality for the difference of Crawford numbers of two linear bounded operators A and B has been obtained. It is shown that (Formula presented.) where c(·) and ω(·) denote the Crawford number and the numerical radius, respectively. The results have been supported by an example. Finally, some applications to operator Hölder functions and operator-functions have been given.
Sharper bounds for the numerical radius of n×n operator matrices
Article, Archiv der Mathematik, 2024, DOI Link
View abstract ⏷
Let A=Aij be an n×n operator matrix, where each Aij is a bounded linear operator on a complex Hilbert space. Among other numerical radius bounds, we show that w(A)≤w(A^), where A^=a^ij is an n×n complex matrix, with (Formula presented.) This is a considerable improvement of the existing bound w(A)≤w(A~), where A~=a~ij is an n×n complex matrix, with (Formula presented.) Further, applying the bounds, we develop the numerical radius bounds for the product of two operators and the commutator of operators. Also, we develop an upper bound for the spectral radius of the sum of the product of n pairs of operators, which improves the existing bound.
Euclidean operator radius inequalities of d-tuple operators and operator matrices
Article, Mathematica Slovaca, 2024, DOI Link
View abstract ⏷
We study Euclidean operator radius inequalities of d-tuple operators as well as the sum and the product of d-tuple operators. A power inequality for the Euclidean operator radius of d-tuple operators is also studied. Further, we study the Euclidean operator radius inequalities of 2 × 2 operator matrices whose entries are d-tuple operators.
A-Davis–Wielandt Radius Bounds of Semi-Hilbertian Space Operators
Guesba M., Barik S., Bhunia P., Paul K.
Article, Bulletin of the Iranian Mathematical Society, 2024, DOI Link
View abstract ⏷
Consider H is a complex Hilbert space and A is a positive operator on H. The mapping ⟨·,·⟩A:H×H→C, defined as y,zA=Ay,z for all y, z∈H, induces a seminorm ·A. The A-Davis–Wielandt radius of an operator S on H is defined as dωAS=supSz,zA2+SzA4:zA=1. We investigate some new bounds for dωAS which refine the existing bounds. We also give some bounds for the 2×2 off-diagonal block matrices.
EUCLIDEAN OPERATOR RADIUS AND NUMERICAL RADIUS INEQUALITIES
Article, Operators and Matrices, 2024, DOI Link
View abstract ⏷
Let T be a bounded linear operator on a complex Hilbert space H. We obtain various lower and upper bounds for the numerical radius of T by developing the Euclidean operator radius bounds of a pair of operators, which are stronger than the existing ones. In particular, we develop an inequality that improves on the inequality Various equality conditions of the existing numerical radius inequalities are also provided. Further, we study the numerical radius inequalities of 2 × 2 off-diagonal operator matrices. Applying the numerical radius bounds of operator matrices, we develop upper bounds of w(T) by using t -Aluthge transform. In particular, we improve the well known inequality where T = |T|1/2|U|T|1/2 is the Aluthge transform of T and T = U|T| is the polar decomposition of T.
Estimates of Euclidean numerical radius for block matrices
Article, Proceedings of the Indian Academy of Sciences: Mathematical Sciences, 2024, DOI Link
View abstract ⏷
We develop several Euclidean numerical radius bounds for the product of two d-tuple operators using positivity criteria of a 2×2 block matrix whose entries are d-tuple operators. From these bounds, by using polar decomposition of operators, we obtain Euclidean numerical radius bounds for d-tuple operators. Among many other bounds, it is shown that (Formula presented.) where we(A) and ‖A‖ are the Euclidean numerical radius and the Euclidean operator norm, respectively, of a d-tuple operator A=(A1,A2,…,Ad). Further, we develop an upper bound for the Euclidean numerical radius of an n×n operator matrix whose entries are d-tuple operators. In particular, it is proved that if [Aij]n×n is an n×n operator matrix then (Formula presented.) where each Aij is a d-tuple operator, 1≤i,j≤n, aij=we(Aij) if i=j, aij=we(|Aji|+|Aij∗|)we(|Aij|+|Aji∗|) if i<j, and aij=0 if i>j. Some related applications are also discussed.
On a New Norm on the Space of Reproducing Kernel Hilbert Space Operators and Berezin Radius Inequalities
Bhunia P., Gurdal M., Paul K., Sen A., Tapdigoglu R.
Article, Numerical Functional Analysis and Optimization, 2023, DOI Link
View abstract ⏷
In this paper, we provide a new norm(α-Berezin norm) on the space of all bounded linear operators defined on a reproducing kernel Hilbert space, which generalizes the Berezin radius and the Berezin norm. We study the basic properties of the α-Berezin norm and develop various inequalities involving the α-Berezin norm. By using the inequalities we obtain various bounds for the Berezin radius of bounded linear operators, which improve on the earlier bounds. Further, we obtain a Berezin radius inequality for the sum of the product of operators, from which we derive new Berezin radius bounds.
NUMERICAL RADIUS INEQUALITIES OF OPERATOR MATRICES FROM A NEW NORM ON B(H)
Bhunia P., Bhanja A., Sain D., Paul K.
Article, Miskolc Mathematical Notes, 2023, DOI Link
View abstract ⏷
This paper is a continuation of a recent work on a new norm, christened the (α,β)- norm, on the space of bounded linear operators on a Hilbert space. We obtain some upper bounds for the said norm of n×n operator matrices. As an application of the present study, we estimate bounds for the numerical radius and the usual operator norm of n×n operator matrices, which generalize the existing ones.
Bounds for the Berezin number of reproducing kernel Hilbert space operators
Article, Filomat, 2023, DOI Link
View abstract ⏷
In this paper, we find new upper bounds for the Berezin number of the product of bounded linear operators defined on reproducing kernel Hilbert spaces. We also obtain some interesting upper bounds concerning one operator, the upper bounds obtained here refine the existing ones. Further, we develop new lower bounds for the Berezin number concerning one operator by using their Cartesian decomposition. In() particular, we prove that ber(A) ≥1√ 2 berℜ(A)±ℑ(A), where ber(A) is the Berezin number of the bounded linear operator A.
A-numerical radius inequalities and A-translatable radii of semi-Hilbert space operators
Guesba M., Bhunia P., Paul K.
Article, Filomat, 2023, DOI Link
View abstract ⏷
We develop A-numerical radius inequalities of the product and the commutator of semi-Hilbert space operators using the notion of A-numerical radius distance and A-seminorm distance. Further, we introduce a pair of translatable radii of semi-Hilbert space operators in the direction of another operator and obtain related inequalities which generalize the relevant inequalities studied in the setting of Hilbert space.
Inequalities Involving Berezin Norm and Berezin Number
Article, Complex Analysis and Operator Theory, 2023, DOI Link
View abstract ⏷
We obtain new inequalities involving Berezin norm and Berezin number of bounded linear operators defined on a reproducing kernel Hilbert space H. Among many inequalities obtained here, it is shown that if A is a positive bounded linear operator on H, then ‖ A‖ ber= ber(A) , where ‖ A‖ ber and ber(A) are the Berezin norm and Berezin number of A, respectively. In contrast to the numerical radius, this equality does not hold for selfadjoint operators, which highlights the necessity of studying Berezin number inequalities independently.
Improvements of A-numerical radius bounds
Nayak R.K., Bhunia P., Paul K.
Article, Hokkaido Mathematical Journal, 2023, DOI Link
View abstract ⏷
We obtain upper and lower bounds for the A-numerical radius inequalities of operators and operator matrices which generalize and improve on the existing ones. We present new upper bounds for the A-numerical radius of the product of two operators. We also develop various inequalities for the A-numerical radius of 2 × 2 operator matrices.
Numerical radius inequalities of 2 × 2 operator matrices
Article, Advances in Operator Theory, 2023, DOI Link
View abstract ⏷
Several upper and lower bounds for the numerical radii of 2 × 2 operator matrices are developed which refine and generalize earlier related bounds. In particular, we show that if B, C are bounded linear operators on a complex Hilbert space, then 12max{‖B‖,‖C‖}+14|‖B+C∗‖-‖B-C∗‖|≤w([0BC0])≤12max{‖B‖,‖C‖}+12max{r12(|B||C∗|),r12(|B∗||C|)},where w(.), r(.) and ‖. ‖ are the numerical radius, spectral radius and operator norm of a bounded linear operator, respectively. We also obtain equality conditions for the numerical radius of the operator matrix [0BC0]. As application of the results obtained, we show that if B, C are normal operators then max{‖B+C‖2,‖B-C‖2}≤∥|B|2+|C|2∥+2w(|B||C|).
Anderson’s theorem and A-spectral radius bounds for semi-Hilbertian space operators
Bhunia P., Kittaneh F., Paul K., Sen A.
Article, Linear Algebra and Its Applications, 2023, DOI Link
View abstract ⏷
Let H be a complex Hilbert space and let A be a positive bounded linear operator on H. Let T be an A-bounded operator on H. For rank(A)=n<∞, we show that if WA(T)⊆D‾(={λ∈C:|λ|≤1}) and WA(T) intersects ∂D(={λ∈C:|λ|=1}) at more than n points, then WA(T)=D‾. In particular, when A is the identity operator on Cn, then this leads to Anderson's theorem in the complex Hilbert space Cn. We introduce the notion of A-compact operators to study analogous result when the space H is infinite dimensional. Further, we develop an upper bound for the A-spectral radius of n×n operator matrices with entries are commuting A-bounded operators, where A=diag(A,A,…,A) is an n×n diagonal operator matrix. Several inequalities involving A-spectral radius of A-bounded operators are also given.
Improved Inequalities for Numerical Radius via Cartesian Decomposition
Bhunia P., Jana S., Moslehian M.S., Paul K.
Article, Functional Analysis and its Applications, 2023, DOI Link
View abstract ⏷
Abstract: We derive various lower bounds for the numerical radius w(A) of a bounded linear operator A defined on a complex Hilbert space, which improve the existing inequality w^2(A)geq frac{1}{4}|A^*A+AA^*| . In particular, for rgeq 1 , we show that tfrac{1}{4}|A^*A+AA^*|leqtfrac{1}{2}(tfrac{1}{2}|operatorname{Re}(A)+operatorname{Im}(A)|^{2r}+tfrac{1}{2}|operatorname{Re}(A)-operatorname{Im}(A)|^{2r})^{1/r} leq w^{2}(A), where operatorname{Re}(A) and operatorname{Im}(A) are the real and imaginary parts of A , respectively. Furthermore, we obtain upper bounds for w^2(A) refining the well-known upper estimate w^2(A)leq frac{1}{2}(w(A^2)+|A|^2) . Criteria for w(A)=frac12|A| and for w(A)=frac{1}{2}sqrt{|A^*A+AA^*|} are also given.
Euclidean Operator Radius Inequalities of a Pair of Bounded Linear Operators and Their Applications
Article, Bulletin of the Brazilian Mathematical Society, 2023, DOI Link
View abstract ⏷
We obtain several sharp lower and upper bounds for the Euclidean operator radius of a pair of bounded linear operators defined on a complex Hilbert space. As applications of these bounds we deduce a chain of new bounds for the classical numerical radius of a bounded linear operator which improve on the existing ones. In particular, we prove that for a bounded linear operator A, 14‖A∗A+AA∗‖+μ2max{‖ℜ(A)‖,‖ℑ(A)‖}≤w2(A)≤w2(|ℜ(A)|+i|ℑ(A)|),where μ= | ‖ ℜ(A) + ℑ(A) ‖ - ‖ ℜ(A) - ℑ(A) ‖ |. This improve the existing upper and lower bounds of the numerical radius, namely, 14‖A∗A+AA∗‖≤w2(A)≤12‖A∗A+AA∗‖.
Numerical radius inequalities for tensor product of operators
Article, Proceedings of the Indian Academy of Sciences: Mathematical Sciences, 2023, DOI Link
View abstract ⏷
The two well-known numerical radius inequalities for the tensor product A⊗ B acting on H⊗ K, where A and B are bounded linear operators defined on complex Hilbert spaces H and K, respectively are 12‖A‖‖B‖≤w(A⊗B)≤‖A‖‖B‖ and w(A) w(B) ≤ w(A⊗ B) ≤ min { w(A) ‖ B‖ , w(B) ‖ A‖ }. In this article, we develop new lower and upper bounds for the numerical radius w(A⊗ B) of the tensor product A⊗ B and study the equality conditions for those bounds.
Numerical radius inequalities of sectorial matrices
Article, Annals of Functional Analysis, 2023, DOI Link
View abstract ⏷
We obtain several upper and lower bounds for the numerical radius of sectorial matrices. We also develop several numerical radius inequalities of the sum, product and commutator of sectorial matrices. The inequalities obtained here are sharper than the existing related inequalities for general matrices. Among many other results we prove that if A is an n× n complex matrix with the numerical range W(A) satisfying W(A)⊆{re±iθ:θ1≤θ≤θ2}, where r> 0 and θ1, θ2∈ [ 0 , π/ 2 ] , then (i)w(A)≥cscγ2‖A‖+cscγ2|‖ℑ(A)‖-‖ℜ(A)‖|,and(ii)w2(A)≥csc2γ4‖AA∗+A∗A‖+csc2γ2|‖ℑ(A)‖2-‖ℜ(A)‖2|, where γ= max { θ2, π/ 2 - θ1} . We also prove that if A, B are sectorial matrices with sectorial index γ∈ [ 0 , π/ 2 ) and they are double commuting, then w(AB) ≤ (1 + sin 2γ) w(A) w(B).
Development of the Berezin Number Inequalities
Article, Acta Mathematica Sinica, English Series, 2023, DOI Link
View abstract ⏷
We present new bounds for the Berezin number inequalities which improve on the existing bounds. We also obtain bounds for the Berezin norm of operators as well as the sum of two operators.
Some New Applications of Berezin Symbols
Bhunia P., Garayev M.T., Paul K., Tapdigoglu R.
Article, Complex Analysis and Operator Theory, 2023, DOI Link
View abstract ⏷
We study some problems of operator theory by using Berezin symbols approach. Namely, we investigate in terms of Berezin symbols invariant subspaces of isometric composition operators on H(Ω). We discuss operator corona problem, in particular, the Toeplitz corona problem. Further, we characterize unitary operators in terms of Berezin symbols. We show that the well known inequality w(A)≥12∥A∥ for numerical radius is not true for the Berezin number of operators, which is defined by ber (A) : = sup λ∈Ω| A~ (λ) | , where A~ (λ) : = 〈 Ak^ λ, k^ λ〉 is the Berezin symbol of operator A: H(Ω) → H(Ω). Finally, we provide a lower bound for ber (A).
Further refinements of davis–wielandt radius inequalities
Bhunia P., Paul K., Barik S.
Article, Operators and Matrices, 2023, DOI Link
View abstract ⏷
Suppose T,S are bounded linear operators on a complex Hilbert space. We show that the Davis-Wielandt radius dw(·) satisfies the following inequalities From the third inequality we obtain the following lower and upper bounds for the Davis-Wielandt radius dw(T) of the operator T: Further, we develop several new lower and upper bounds for the Davis-Wielandt radius of the operator T which improve the existing ones. Application of these bounds are also provided. Mathematics subject classification (2020): 47A12, 47A30, 15A60, 47A50.
Davis–Wielandt–Berezin radius inequalities of reproducing kernel Hilbert space operators
Article, Afrika Matematika, 2023, DOI Link
View abstract ⏷
Several upper and lower bounds of the Davis–Wielandt–Berezin radius of bounded linear operators defined on a reproducing kernel Hilbert space are given. Further, an inequality involving the Berezin number and the Davis–Wielandt–Berezin radius for the sum of two bounded linear operators is obtained, namely, if A and B are reproducing kernel Hilbert space operators, then η(A+B)≤η(A)+η(B)+ber(A∗B+B∗A), where η(·) and ber(·) are the Davis–Wielandt–Berezin radius and the Berezin number, respectively.
Numerical radius inequalities and estimation of zeros of polynomials
Article, Georgian Mathematical Journal, 2023, DOI Link
View abstract ⏷
Let A be a bounded linear operator defined on a complex Hilbert space and let | A | = (A A) 1 2 {|A|=(A^{∗}A)^{frac{1}{2}}}. Among other refinements of the well-known numerical radius inequality w 2(A) ≤ 1 2∥A A + AA ∗ ∥, we show that w 2(A) ≤ 1 4w 2(| A| + i |A ∗ |) + 1 8∥| A | 2 + | A ∗ | 2 ∥+ 1 4w(| A || A ∗ |) ≤ 1 2∥A A + AA ∗ ∥. w. Also, we develop inequalities involving the numerical radius and the spectral radius for the sum of the product operators, from which we derive the inequalities w p(A) ≤ 1 2w(| A | p + i| A ∗ | p) ≤ ∥A ∥p w^{p}(A)leqfrac{1}{sqrt{2}}w(|A|^{p}+{rm i}|A^{∗}|^{p})leq|A|^{p} for all p ≥ 1 {pgeq 1}. Further, we derive new bounds for the zeros of complex polynomials.
A-numerical radius: New inequalities and characterization of equalities
Article, Hacettepe Journal of Mathematics and Statistics, 2023, DOI Link
View abstract ⏷
We develop new lower bounds for the A-numerical radius of semi-Hilbertian space oper-ators, and applying these bounds we obtain upper bounds for the A-numerical radius of the commutators of operators. The bounds obtained here improve on the existing ones. Further, we provide characterizations for the equality of the existing A-numerical radius inequalities of semi-Hilbertian space operators.
Refinement of numerical radius inequalities of complex Hilbert space operators
Article, Acta Scientiarum Mathematicarum, 2023, DOI Link
View abstract ⏷
We develop upper and lower bounds for the numerical radius of 2 × 2 off-diagonal operator matrices, which generalize and improve on some existing ones. We also show that if A is a bounded linear operator on a complex Hilbert space, then for all r≥ 1 , w2r(A)≤14‖|A|2r+|A∗|2r‖+12min{‖ℜ(|A|r|A∗|r)‖,wr(A2)}where w(A), ‖ A‖ and ℜ(A) , respectively, stand for the numerical radius, the operator norm and the real part of A. This (for r= 1) improves on some existing well-known numerical radius inequalities.
Corrigendum to “Development of inequalities and characterization of equality conditions for the numerical radius” [Linear Algebra Appl. 630 (2021) 306–315, (S0024379521003116), (10.1016/j.laa.2021.08.014)]
Erratum, Linear Algebra and Its Applications, 2023, DOI Link
View abstract ⏷
The purpose of this article is to present a correct version of [1, Lemma 2.13] and [1, Th. 2.14].
A-Numerical Radius Inequalities in Semi-Hilbertian Spaces
Bhunia P., Dragomir S.S., Moslehian M.S., Paul K.
Book chapter, Infosys Science Foundation Series in Mathematical Sciences, 2022, DOI Link
View abstract ⏷
Over the years, many mathematicians have studied different generalizations of the usual numerical radius of a bounded linear operator acting on a complex Hilbert space (H), see [2, 147, 174].
Numerical Radius of Operator Matrices and Applications
Bhunia P., Dragomir S.S., Moslehian M.S., Paul K.
Book chapter, Infosys Science Foundation Series in Mathematical Sciences, 2022, DOI Link
View abstract ⏷
Suppose (H) is a complex Hilbert space, and T is a bounded linear operator on (H).
p-Numerical Radius Inequalities of an n-Tuple of Operators
Bhunia P., Dragomir S.S., Moslehian M.S., Paul K.
Book chapter, Infosys Science Foundation Series in Mathematical Sciences, 2022, DOI Link
View abstract ⏷
Let (formula presented) be an n-tuple of operators in (formula presented). The joint numerical range of (formula presented) is defined by (formula presented).
Numerical Radius Inequalities of Product of Operators
Bhunia P., Dragomir S.S., Moslehian M.S., Paul K.
Book chapter, Infosys Science Foundation Series in Mathematical Sciences, 2022, DOI Link
View abstract ⏷
The spectral mapping theorem ensures that for a bounded linear operator A on a complex Hilbert space (formula presented) where f is an analytic function on a domain containing (A) Unfortunately, there is no such relation for the numerical range of a bounded linear operator, that is, (formula presented).
Fundamental Numerical Radius Inequalities
Bhunia P., Dragomir S.S., Moslehian M.S., Paul K.
Book chapter, Infosys Science Foundation Series in Mathematical Sciences, 2022, DOI Link
View abstract ⏷
The concept of a numerical range is a natural extension of quadratic forms studied in linear algebra. To be more precise, the numerical range of a bounded linear operator A on a complex Hilbert space (H), to be denoted by W(A), is defined as the range of the continuous mapping (formula presented) defined on the unit sphere of the Hilbert space (H), that is, The study of numerical range assists in understanding the behavior of a bounded linear operator.
Lectures on Numerical Radius Inequalities
Bhunia P., Dragomir S.S., Moslehian M.S., Paul K.
Book chapter, Infosys Science Foundation Series in Mathematical Sciences, 2022,
Preliminaries
Bhunia P., Dragomir S.S., Moslehian M.S., Paul K.
Book chapter, Infosys Science Foundation Series in Mathematical Sciences, 2022, DOI Link
View abstract ⏷
In this chapter, we collect some basic facts needed to study the numerical range and numerical radius of a bounded linear operator defined on a Hilbert space and fix our notation.
Preface
Bhunia P., Dragomir S.S., Moslehian M.S., Paul K.
Editorial, Infosys Science Foundation Series in Mathematical Sciences, 2022, DOI Link
Research Problems
Bhunia P., Dragomir S.S., Moslehian M.S., Paul K.
Book chapter, Infosys Science Foundation Series in Mathematical Sciences, 2022, DOI Link
View abstract ⏷
In this chapter, we present a number of research problems related to numerical range and numerical radius with various levels of difficulty. For most problems, we provide some references helping the reader to see the background needed to well understand and start thinking about them. Some of the problems given below are known and some are new.
Operator Space Numerical Radius of $$2times 2$$ Block Matrices
Bhunia P., Dragomir S.S., Moslehian M.S., Paul K.
Book chapter, Infosys Science Foundation Series in Mathematical Sciences, 2022, DOI Link
View abstract ⏷
In this chapter, following [57], the notion of complete numerical radius norm is studied and it is shown that the complete numerical radius norm of a completely bounded homomorphism can be computed in terms of the completely bounded norm of the map.
Bounds of the Numerical Radius Using Buzano’s Inequality
Bhunia P., Dragomir S.S., Moslehian M.S., Paul K.
Book chapter, Infosys Science Foundation Series in Mathematical Sciences, 2022, DOI Link
View abstract ⏷
One of the most fundamental and widely used inequalities in mathematics is the celebrated Cauchy–Schwarz inequality, also known as the Cauchy–Bunyakovsky–Schwarz inequality. The elementary form of Cauchy–Schwarz inequality states that if (formula presented) and (formula presented) are real numbers, then (formula presented) Its general form in an inner-product space is (1.2 ). The Cauchy–Schwarz inequality was wonderfully refined in 1971 by Buzano [49].
Numerical Radius Inequalities for Products and Sums of Semi-Hilbertian Space Operators
Article, Filomat, 2022, DOI Link
View abstract ⏷
New inequalities for the A-numerical radius of the products and sums of operators acting on a semi-Hilbert space, i.e. a space generated by a positive semidefinite operator A, are established. In particular, for every operators T and S which admit A-adjoints, it is proved that.
Some improvements of numerical radius inequalities of operators and operator matrices
Article, Linear and Multilinear Algebra, 2022, DOI Link
View abstract ⏷
We obtain upper bounds for the numerical radius of a product of Hilbert space operators which improve on the existing upper bounds. We generalize the numerical radius inequalities of (Formula presented.) operator matrices by using non-negative continuous functions on (Formula presented.). We also obtain some upper and lower bounds for the B-numerical radius of operator matrices, where B is the diagonal operator matrix whose each diagonal entry is a positive operator A. We show that these bounds generalize and improve on the existing bounds.
Annular bounds for the zeros of a polynomial from companion matrices
Article, Advances in Operator Theory, 2022, DOI Link
View abstract ⏷
Let p(z) = zn+ an-1zn-1+ an-2zn-2+ ⋯ + a1z+ a be a complex polynomial with a≠ 0 and n≥ 3. Several new upper bounds for the moduli of the zeros of p are developed. In particular, if α=∑j=0n-1|aj|2 and z is any zero of p, then we show that |z|2≤cos2πn+1+|an-2|+14(|an-1|+α)2+12α2-|an-1|2+12α,which is sharper than the existing bound, given as, |z|2≤cos2πn+1+14(|an-1|+α)2+α,if and only if 2|an-2|<∑j=0n-1|aj|2-∑j=0n-2|aj|2. The upper bounds obtained here enable us to describe smaller annuli in the complex plane containing all the zeros of p.
BEREZIN NUMBER INEQUALITIES OF OPERATORS ON REPRODUCING KERNEL HILBERT SPACES
Article, Rocky Mountain Journal of Mathematics, 2022, DOI Link
View abstract ⏷
Several new upper bounds for the Berezin number of bounded linear operators defined on reproducing kernel Hilbert spaces are given. The bounds obtained here improve on the earlier ones.
Refinement of seminorm and numerical radius inequalities of semi-Hilbertian space operators
Bhunia P., Nayak R.k., Paul K.
Article, Mathematica Slovaca, 2022, DOI Link
View abstract ⏷
Let H be a complex Hilbert space and A be a non-zero positive bounded linear operator on H. The main aim of this paper is to discuss a general method to develop A-operator seminorm and A-numerical radius inequalities of semi-Hilbertian space operators using the existing corresponding inequalities of bounded linear operators on H. Among many other inequalities we prove that if S, T, X ∈ BA (H), i.e., if A-adjoint of S, T, X exist, then 2∥S ]A XT∥A ≤ ∥SS]A X + XT T]A ∥A. Further, we prove that if T ∈ BA(H), then 1 4 ∥T ]A T + T T]A ∥A ≤ 1 8 ∥T + T ]A ∥ 2 A + ∥T - T ]A ∥ 2 A ≤ 1 8 ∥T + T ]A ∥ 2 A + ∥T - T ]A ∥ 2 A + 1 8 c 2 A.
On the Davis-Wielandt shell of an operator and the Davis-Wielandt index of a normed linear space
Article, Collectanea Mathematica, 2022, DOI Link
View abstract ⏷
We study the Davis-Wielandt shell and the Davis-Wielandt radius of an operator on a normed linear space X. We show that after a suitable modification, the modified Davis-Wielandt radius defines a norm on L(X) which is equivalent to the usual operator norm on L(X). We introduce the Davis-Wielandt index of a normed linear space and compute its value explicitly in case of some particular polyhedral Banach spaces. We also present a general method to estimate the Davis-Wielandt index of any polyhedral finite-dimensional Banach space.
Generalized A-Numerical Radius of Operators and Related Inequalities
Article, Bulletin of the Iranian Mathematical Society, 2022, DOI Link
View abstract ⏷
Let H be a complex Hilbert space with inner product ⟨ · , · ⟩ and let A be a non-zero bounded positive linear operator on H. Let BA(H) denote the algebra of all bounded linear operators on H which admit A-adjoint, and let NA(·) be a seminorm on BA(H). The generalized A-numerical radius of T∈ BA(H) is defined as ωNA(T)=supθ∈RNA(eiθT+e-iθT♯A2),where T♯A stands for a distinguished A-adjoint of T. In this article, we focus on the development of several generalized A-numerical radius inequalities. We also develop bounds for the generalized A-numerical radius of sum and product of operators.
On Numerical Radius and Crawford Number Attainment Sets of a Bounded Linear Operator
Sain D., Mal A., Bhunia P., Paul K.
Review, Journal of Convex Analysis, 2021,
View abstract ⏷
We completely characterize the Crawford number attainment set and the numerical radius attainment set of a bounded linear operator on a Hilbert space. We study the intersection properties of the corresponding attainment sets of numerical radius, Crawford number, norm, minimum norm of a bounded linear operator defined on a normed space. Our study illustrates the similarities and the differences of the extremal properties of a bounded linear operator on a Hilbert space and a general normed space.
Estimations of Zeros of a Polynomial Using Numerical Radius Inequalities
Bhunia P., Bag S., Nayak R.K., Paul K.
Article, Kyungpook Mathematical Journal, 2021, DOI Link
View abstract ⏷
We present new bounds for the numerical radius of bounded linear operators and 2 x 2 operator matrices. We apply upper bounds for the numerical radius to the Frobenius companion matrix of a complex monic polynomial to obtain new estimations for the zeros of that polynomial. We also show with numerical examples that our new estimations improve on the existing estimations.
Bounds for the Davis–Wielandt radius of bounded linear operators
Bhunia P., Bhanja A., Bag S., Paul K.
Article, Annals of Functional Analysis, 2021, DOI Link
View abstract ⏷
We obtain upper and lower bounds for the Davis–Wielandt radius of bounded linear operators defined on a complex Hilbert space, which improve on the existing ones. We also obtain bounds for the Davis–Wielandt radius of operator matrices. We determine the exact value of the Davis–Wielandt radius of some special type of operator matrices.
Sharp inequalities for the numerical radius of hilbert space operators and operator matrices
Bhunia P., Paul K., Nayak R.K.
Article, Mathematical Inequalities and Applications, 2021, DOI Link
View abstract ⏷
We present new upper and lower bounds for the numerical radius of a bounded linear operator defined on a complex Hilbert space, which improve on the existing bounds. Among many other inequalities proved in this article, we show that for a non-zero bounded linear operator T on a complex Hilbert space H, w(T) ≥ ||T2 || + m2||(TT2 || ) , where w(T) is the numerical radius of T and m(T2) is the Crawford number of T2 . This substantially improves on the existing inequality w(T) ≥ ||T2 ||. We also obtain some upper and lower bounds for the numerical radius of operator matrices and illustrate with numerical examples that these bounds are better than the existing bounds.
Numerical radius inequalities of operator matrices with applications
Article, Linear and Multilinear Algebra, 2021, DOI Link
View abstract ⏷
We present upper and lower bounds for the numerical radius of (Formula presented.) operator matrices which improve on the existing bounds for the same. As an application of the results obtained we give a better estimation for the zeros of a polynomial.
New upper bounds for the numerical radius of Hilbert space operators
Article, Bulletin des Sciences Mathematiques, 2021, DOI Link
View abstract ⏷
In this paper we present new upper bounds for the numerical radius of bounded linear operators defined on a complex Hilbert space. Further we obtain estimations for upper bounds for the numerical radius of the sum of the product of bounded linear operators. We show that the bounds obtained here improve on the existing well-known upper bounds.
A-Numerical Radius Orthogonality and Parallelism of Semi-Hilbertian Space Operators and Their Applications
Article, Bulletin of the Iranian Mathematical Society, 2021, DOI Link
View abstract ⏷
In this paper, we aim to introduce and characterize the numerical radius orthogonality of operators on a complex Hilbert space H which are bounded with respect to the seminorm induced by a positive operator A on H. Moreover, a characterization of the A-numerical radius parallelism for A-rank one operators is obtained. As applications of the results obtained, we derive some A-numerical radius inequalities of operator matrices, where A is the operator diagonal matrix whose each diagonal entry is a positive operator A on a complex Hilbert space H.
Correction to: A-Numerical Radius Orthogonality and Parallelism of Semi-Hilbertian Space Operators and Their Applications (Bulletin of the Iranian Mathematical Society, (2021), 47, 2, (435-457), 10.1007/s41980-020-00392-8)
Erratum, Bulletin of the Iranian Mathematical Society, 2021, DOI Link
View abstract ⏷
In the original article published, during the final typesetting stage the equation in the proof of the Theorem 3.5 was published incorrectly. The correct equation is: (Formula presented.).
Bounds for zeros of a polynomial using numerical radius of Hilbert space operators
Article, Annals of Functional Analysis, 2021, DOI Link
View abstract ⏷
We obtain bounds for the numerical radius of 2 × 2 operator matrices which improve on the existing bounds. We also show that the inequalities obtained here generalize the existing ones. As an application of the results obtained here, we estimate the bounds for the zeros of a monic polynomial and illustrate with numerical examples that the bounds are better than the existing ones.
Improvement of A-Numerical Radius Inequalities of Semi-Hilbertian Space Operators
Bhunia P., Nayak R.K., Paul K.
Article, Results in Mathematics, 2021, DOI Link
View abstract ⏷
Let H be a complex Hilbert space and let A be a positive operator on H. We obtain new bounds for the A-numerical radius of operators in semi-Hilbertian space BA(H) that generalize and improve on the existing ones. Further, we estimate an upper bound for the A-operator seminorm of 2 × 2 operator matrices, where A=diag(A,A). The bound obtained here generalizes the earlier related bound.
On a new norm on B(H) and its applications to numerical radius inequalities
Sain D., Bhunia P., Bhanja A., Paul K.
Article, Annals of Functional Analysis, 2021, DOI Link
View abstract ⏷
We introduce a new norm on the space of all bounded linear operators on a complex Hilbert space, which generalizes the numerical radius norm, the usual operator norm and the modified Davis–Wielandt radius norm. We study basic properties of this norm, including the upper and the lower bounds for it. As an application of the present study, we estimate bounds for the numerical radius of bounded linear operators. We illustrate that our results improve on some of the important existing numerical radius inequalities. Other application of this new norm have also studied.
New Inequalities for Davis–Wielandt Radius of Hilbert Space Operators
Bhunia P., Bhanja A., Paul K.
Article, Bulletin of the Malaysian Mathematical Sciences Society, 2021, DOI Link
View abstract ⏷
Let T be a bounded linear operator on a complex Hilbert space and d w(T) denote the Davis–Wielandt radius of the operator T. We prove that dw2(T)≤min0≤α≤1‖α|T|2+(1-α)|T∗|2+|T|4‖and dw2(T)≤min0≤α≤1‖α|T|2+(1-α)|T∗|2+|T∗|4‖,where |T|=T∗T,|T∗|=TT∗. We also develop several other bounds for the Davis–Wielandt radius and prove that the bounds obtained here are better than the existing ones.
Furtherance of numerical radius inequalities of Hilbert space operators
Article, Archiv der Mathematik, 2021, DOI Link
View abstract ⏷
If A, B are bounded linear operators on a complex Hilbert space, then we prove that w(A)≤12(‖A‖+r(|A||A∗|)),w(AB±BA)≤22‖B‖w2(A)-c2(R(A))+c2(I(A))2,where w(·) , ∥ · ∥ , and r(·) are the numerical radius, the operator norm, the Crawford number, and the spectral radius respectively, and R(A) , I(A) are the real part, the imaginary part of A respectively. The inequalities obtained here generalize and improve on the existing well known inequalities.
ON GENERALIZED DAVIS–WIELANDT RADIUS INEQUALITIES OF SEMI–HILBERTIAN SPACE OPERATORS
Bhanja A., Bhunia P., Paul K.
Article, Operators and Matrices, 2021, DOI Link
View abstract ⏷
Let) A be a positive (semidefinite) operator on a complex Hilbert space H and let (Formula presented). We obtain upper and lower bounds for the A-Davis-Wielandt radius of semiOA Hilbertian space operators, which generalize and improve on the existing ones. Further, we derive upper bounds for the A-Davis-Wielandt radius of the sum of the product of semi-Hilbertian space operators. We also obtain upper bounds for the A-Davis-Wielandt radius of 2×2 operator matrices. (Finally,) we determine the exact value for the A-Davis-Wielandt radius of two operator matrices (Formula presented) and (Formula presented), where X is a semi-Hilbertian space operator, and I, O are OO OO the identity operator, the zero operator on H, respectively.
Development of inequalities and characterization of equality conditions for the numerical radius
Article, Linear Algebra and Its Applications, 2021, DOI Link
View abstract ⏷
Let A be a bounded linear operator on a complex Hilbert space and ℜ(A) (ℑ(A)) denote the real part (imaginary part) of A. Among other refinements of the lower bounds for the numerical radius of A, we prove that [Formula presented] where w(A) and ‖A‖ are the numerical radius and operator norm of A, respectively. We study the equality conditions for [Formula presented] and prove that [Formula presented] if and only if the numerical range of A is a circular disk with center at the origin and radius [Formula presented]. We also obtain upper bounds for the numerical radius of commutators of operators which improve on the existing ones.
Proper Improvement of Well-Known Numerical Radius Inequalities and Their Applications
Article, Results in Mathematics, 2021, DOI Link
View abstract ⏷
New inequalities for the numerical radius of bounded linear operators defined on a complex Hilbert space H are given. In particular, it is established that if T is a bounded linear operator on a Hilbert space H then w2(T)≤min0≤α≤1∥αT∗T+(1-α)TT∗∥,where w(T) is the numerical radius of T. The inequalities obtained here are non-trivial improvement of the well-known numerical radius inequalities. As an application we estimate bounds for the zeros of a complex monic polynomial.
REFINEMENTS OF NORM AND NUMERICAL RADIUS INEQUALITIES
Article, Rocky Mountain Journal of Mathematics, 2021, DOI Link
View abstract ⏷
Several refinements of norm and numerical radius inequalities of bounded linear operators on a complex Hilbert space are given. In particular, we show that if A is a bounded linear operator on a complex Hilbert space, then (equation presented) where k k, w.( ) and c( ) are the operator norm, the numerical radius and the Crawford number, respectively. Further, we prove that if A; D are bounded linear operators on a complex Hilbert space, then (equation presented) where jAj2 = A*A and jDj2 = D*D. This is a refinement of a well-known inequality obtained by Bhatia and Kittaneh.
On inequalities for a-numerical radius of operators
Bhunia P., Paul K., Nayak R.K.
Article, Electronic Journal of Linear Algebra, 2020, DOI Link
View abstract ⏷
Let A be a positive operator on a complex Hilbert space H. Inequalities are presented concerning upper and lower bounds for A-numerical radius of operators, which improve on and generalize the existing ones, studied recently in [A. Zamani. A-Numerical radius inequalities for semi-Hilbertian space operators. Linear Algebra Appl., 578:159–183, 2019.]. Also, some inequalities are obtained for B-numerical radius of 2 × 2 operator matrices, where B is the 2 × 2 diagonal operator matrix whose diagonal entries are A. Further, upper bounds are obtained for A-numerical radius for product of operators, which improve on the existing bounds.
Bounds of numerical radius of bounded linear operators using t–Aluthge transform
Article, Mathematical Inequalities and Applications, 2020, DOI Link
View abstract ⏷
We develop a number of inequalities to obtain bounds for the numerical radius of a bounded linear operator defined on a complex Hilbert space using the properties of t -Aluthge transform. We show that the bounds obtained are sharper than the existing bounds.
Refinements of A-numerical radius inequalities and their applications
Bhunia P., Nayak R.K., Paul K.
Article, Advances in Operator Theory, 2020, DOI Link
View abstract ⏷
We present sharp lower bounds for the A-numerical radius of semi-Hilbertian space operators. We also present an upper bound. Further we compute new upper bounds for the B-numerical radius of 2 × 2 operator matrices where B= diag(A, A) , A being a positive operator. As an application of the A-numerical radius inequalities, we obtain a bound for the zeros of a polynomial which is quite a bit improvement of some famous existing bounds for the zeros of polynomials.
Bounds for eigenvalues of the adjacency matrix of a graph
Article, Journal of Interdisciplinary Mathematics, 2019, DOI Link
View abstract ⏷
We obtain bounds for the largest and least eigenvalues of the adjacency matrix of a simple undirected graph. We find upper bound for the second largest eigenvalue of the adjacency matrix. We prove that the bounds obtained here improve on the existing bounds and also illustrate them with examples.
Numerical radius inequalities and its applications in estimation of zeros of polynomials
Article, Linear Algebra and Its Applications, 2019, DOI Link
View abstract ⏷
We present some upper and lower bounds for the numerical radius of a bounded linear operator defined on complex Hilbert space, which improves on the existing upper and lower bounds. We also present an upper bound for the spectral radius of sum of product of n pairs of operators. As an application of the results obtained, we provide a better estimation for the zeros of a given polynomial.
On the numerical index of polyhedral Banach spaces
Sain D., Paul K., Bhunia P., Bag S.
Article, Linear Algebra and Its Applications, 2019, DOI Link
View abstract ⏷
The computation of the numerical index of a Banach space is an intriguing problem, even in case of two-dimensional real polyhedral Banach spaces. In this article we present a general method to estimate the numerical index of any finite-dimensional real polyhedral Banach space, by considering the action of only finitely many functionals, on the unit sphere of the space. We further obtain the exact numerical index of a family of 3-dimensional polyhedral Banach spaces for the first time, in order to illustrate the applicability of our method.
