A note on operator norm inequalities

Banach Journal of Mathematical Analysis, 2012

In the present paper, taking some advantages offered by the context of finite dimensional Hilbert spaces, we shall give a complete characterizations of certain distinguished classes of operators (self-adjoint, unitary reflection, normal) in terms of operator inequalities. These results extend previous characterizations obtained by the second author.

Linear and Multilinear Algebra, 2011

, Elsner, Hershkowitz and Pinkus characterized functions F : IR n + →

Linear and Multilinear Algebra, 2004

W 0 ðR A, B J j Þ where VðÁÞ is the joint spatial numerical range, W 0 ðÁÞ is the algebraic numerical range and J is a norm ideal of BðEÞ: We shall show that this inclusion becomes an equality when R A, B is taken to be a derivation. Also, we deduce that wðU A, B J j Þ ! 2ð ffiffi ffi 2 p À 1ÞwðAÞwðBÞ, for A, B 2 BðEÞ and J is a norm ideal of BðEÞ, where wðÁÞ is the numerical radius. On the other hand, in the particular case when E is a Hilbert space, we shall prove that the lower estimate bound kU A, B jJk ! 2ð ffiffi ffi 2 p À 1ÞkAkkBk holds, if one of the following two conditions is satisfied: (i) J is a standard operator algebra of BðEÞ and A, B 2 J: (ii) J is a norm ideal of BðEÞ and A, B 2 BðEÞ:

Linear Algebra and its Applications, 1990

Let F be a surjective linear mapping between the algebras L(H) and L(K) of all bounded operators on nontrivial complex Hilbert spaces H and K respectively. For any positive integer k let W,(A) denote the kth numerical range of an operator A on H. If k is strictly less than one-half the dimension of H and W,(F(A)) = Wk. A) for ah A from L(H), then there is a unitary mapping U: H + K such that either F(A) = UAu* or F(A) = (UAU*)' for every A E L(H), where the transposition is taken in any basis of K, fixed in advance. This generalizes the result of S. Pierce and W. Watkins on finite-dimensional spaces. The case of k greater than or equal to one-half of the dimension of H is also treated using our method. Our proofs depend on a characterization of those linear operators preserving projections of rank one, which is of independent interest.

Linear Algebra and its Applications, 2009

We give an extension of Hua's inequality in pre-Hilbert C * -modules without using convexity or the classical Hua's inequality. As a consequence, some known and new generalizations of this inequality are deduced. Providing a Jensen inequality in the content of Hilbert C * -modules, another extension of Hua's inequality is obtained. We also present an operator Hua's inequality, which is equivalent to operator convexity of given continuous real function. G.-S. Yang and B.-K. Han extended this result for a finite sequence of complex numbers. C.E.M. Pearce and J.E. Pečarić [14] generalized Hua's inequality for real convex functions; see also [1]. S.S. Dragomir and G.-S. Yang [2] extended Hua's inequality in the setting of real inner product spaces by applying Hua's inequality for n = 1. Their result was generalized by 2000 Mathematics Subject Classification. Primary 47A63; secondary 46L08, 47B10, 47A30, 47B15, 26D07, 15A60.

2017

Let be a Hilbert space equipped with the inner product , and let be the algebra of bounded linear operators acting on . We recall that the numerical range (also known as the field of values) of is the collection of all complex numbers of the form where is a unit vector in . i.e. See, ([2], [5], [8]) which is useful for studying operators. In particular, the geometrical properties of the numerical range often provide useful information about the algebraic and analytic properties of the operator . For instance, if and only if ; is real if and only if , has no interior points if and only if there are complex numbers, and with such that is self-adjoint. Moreover, the closure of denoted by , always contains the spectrum of denoted by . See, [8] Let denote the set of compact operators on and be the canonical quotient map. The essential numerical range of , denoted by is the set; See, ([1], [2], [3]) where the intersection runs over the compact operators . Chacon and Chacon [3] gave some o...

Georgian Mathematical Journal, 2019

In this paper, we show some refinements of generalized numerical radius inequalities involving the Young and Heinz inequality. In particular, we present w_{p}^{p}(A_{1}^{*}T_{1}B_{1},\dots,A_{n}^{*}T_{n}B_{n})\leq\frac{n^{1-\frac{1% }{r}}}{2^{\frac{1}{r}}}\bigg{\|}\sum_{i=1}^{n}[B_{i}^{*}f^{2}(|T_{i}|)B_{i}]^{% rp}+[A_{i}^{*}g^{2}(|T_{i}^{*}|)A_{i}]^{rp}\bigg{\|}^{\frac{1}{r}}-\inf_{\|x\|% =1}\eta(x), where {T_{i},A_{i},B_{i}\in\mathbb{B}(\mathscr{H})} {(1\leq i\leq n)} , f and g are nonnegative continuous functions on {[0,\infty)} satisfying {f(t)g(t)=t} for all {t\in[0,\infty)} , {p,r\geq 1} , {N\in\mathbb{N}} , and \displaystyle\eta(x)=\frac{1}{2}\sum_{i=1}^{n}\sum_{j=1}^{N}\Bigl{(}\sqrt[2^{j% }]{\big{\langle}(A_{i}^{*}g^{2}(|T_{i}^{*}|)A_{i})^{p}x,x\big{\rangle}^{2^{j-1% }-k_{j}}\big{\langle}(B_{i}^{*}f^{2}(|T_{i}|)B_{i})^{p}x,x\big{\rangle}^{k_{j}}} \displaystyle -\sqrt[2^{j}]{\big{\langle}(B_{i}^{*}f^{2}(|T_{i}|)B_{i}% )^{p}x,x\big{\rangle}^{k_{j}+1}\big{\langle}(A_{i}^{*}g^{2...

Proceedings of the American Mathematical Society, 2005

We prove several spectral radius inequalities for sums, products, and commutators of Hilbert space operators. Pinching inequalities for the spectral radius are also obtained.

2021

Consider a complex Hilbert space (H, 〈·, ·〉). Let B (H) denote the algebra of all bounded linear operators acting on (H, 〈·, ·〉) An operator A is said to be positive (denoted by A ≥ 0) if 〈Ax, x〉 ≥ 0 for all x ∈ H, and also an operator A is said to be strictly positive (denoted by A > 0) if A is positive and invertible. The Gelfand map f (t) 7→ f (A) is an isometrically ∗isomorphism between the C-algebra C (sp (A)) of continuous functions on the spectrum sp (A) of a self-adjoint operator A and the C-algebra generated by 1H and A. If f, g ∈ C (sp (A)), then f (t) ≥ g (t) (t ∈ sp (A)) implies that f (A) ≥ g (A). On the other hand, when A ∈ B(H) is such that RA > 0, then A is said to be accretive. When H is finite dimensional, we identify B(H) with the algebra Mn of all complex n × n matrices. Given a matrix monotone function f : (0,∞) → (0,∞) with f(1) = 1, and two accretive matrices A,B, there is a matrix mean associated with f , denoted by σf or σ, defined by [1]

Journal of the Egyptian Mathematical Society, 2012