Solution of Inequalities in the theory of operator

Mathematical Physics, Analysis and Geometry

Let $H:\text {dom}(H)\subseteq \mathfrak {F}\to \mathfrak {F}$ H : dom ( H ) ⊆ F → F be self-adjoint and let $A:\text {dom}(H)\to \mathfrak {F}$ A : dom ( H ) → F (playing the role of the annihilation operator) be H-bounded. Assuming some additional hypotheses on A (so that the creation operator A∗ is a singular perturbation of H), by a twofold application of a resolvent Kreı̆n-type formula, we build self-adjoint realizations $\widehat H$ H ̂ of the formal Hamiltonian H + A∗ + A with $\text {dom}(H)\cap \text {dom}(\widehat H)=\{0\}$ dom ( H ) ∩ dom ( H ̂ ) = { 0 } . We give an explicit characterization of $\text {dom}(\widehat H)$ dom ( H ̂ ) and provide a formula for the resolvent difference $(-\widehat H+z)^{-1}-(-H+z)^{-1}$ ( − H ̂ + z ) − 1 − ( − H + z ) − 1 . Moreover, we consider the problem of the description of $\widehat H$ H ̂ as a (norm resolvent) limit of sequences of the kind $H+A^{*}_{n}+A_{n}+E_{n}$ H + A n ∗ + A n + E n , where the An’s are regularized operators appr...

arXiv (Cornell University), 2020

We show that a densely defined closable operator A such that the resolvent set of A 2 is not empty, is necessarily closed. This result is then extended to the case of a polynomial p(A). We also generalize a recent result by Sebestyén-Tarcsay concerning the converse of a result by J. von Neumann. Other interesting consequences are also given, one of them being a proof that if T is a quasinormal (unbounded) operator such that T n is normal for some n ≥ 2, then T is normal. By a recent result by Pietrzycki-Stochel, we infer that a closed subnormal operator such that T n is normal, must be normal. Another remarkable result is the fact that a hyponormal operator A, bounded or not, such that A p and A q are self-adjoint for some co-prime numbers p and q, is self-adjoint. It is also shown that an invertible operator (bounded or not) A for which A p and A q are normal for some co-prime numbers p and q, is normal. These two results are shown using Bézout's theorem in arithmetic. Notation First, we assume that readers have some familiarity with the standard notions and results in operator theory (see e.g. [17] and [25] for some background). We do recall most of the needed notions though. First, note that in this paper all operators are linear. Let H be a complex Hilbert space and let B(H) be the algebra of all bounded linear operators defined from H into H. If S and T are two linear operators with domains D(S) ⊂ H and D(T) ⊂ H respectively, then T is said to be an extension of S, written S ⊂ T , when D(S) ⊂ D(T) and S and T coincide on D(S). The product ST and the sum S + T of two operators S and T are defined in the usual fashion on the natural domains: D(ST) = {x ∈ D(T) : T x ∈ D(S)} and D(S + T) = D(S) ∩ D(T).

Integral Equations and Operator Theory, 1986

arXiv: Functional Analysis, 2020

In this survey, we shall present characterizations of some distinguished classes of Hilbertian bounded linear operators (namely, normal operators, selfadjoint operators, and unitary operators) in terms of operator inequalities related to the arithmetic-geometric mean inequality. For the class of all normal operators, we shall present new general characterizations.

Linear and Multilinear Algebra, 2011

Let BðH Þ, IðH Þ and UðH Þ be the C Ã-algebra of all bounded linear operators acting on a complex Hilbert space H, the set of all invertible elements in BðH Þ and the class of all unitary operators in BðH Þ, respectively. In this note, we shall show that if S 2 IðH Þ, then the injective norm of S S À1 þ S À1 S in the tensor product space BðH Þ BðH Þ attains its minimal value 2 if and only if S is normal and satisfies the condition j þ j 2 for every , in the spectrum (S) of S. Finally, it is shown that if S 2 IðH Þ, then the inequality kSXS À1 þ S À1 XSk 2kXk holds for all X in BðH Þ if and only if S 2 R Ã UðH Þ.

Linear Algebra and its Applications, 2006

Let T be a bounded linear operator on a complex Hilbert space H. In this paper we introduce the class, denoted QA, of operators satisfying T * |T 2 |T T * |T | 2 T and we prove basic structural properties of these operators. Using these results, we also prove that if E is the Riesz idempotent for a non-zero isolated point λ 0 of the spectrum of T ∈ QA, then E is self-adjoint, and we give a necessary and sufficient condition for T ⊗ S to be in QA when T and S are both non-zero operators.

Linear Algebra and its Applications, 2004

Let B(H) be the C *-algebra of all bounded linear operators on a complex Hilbert space H, S be an invertible and selfadjoint operator in B(H) and let (I,. I) denote a norm ideal of B(H). In this note, we shall show the following inequality:

Filomat, 2019

We prove several numerical radius inequalities for products of two Hilbert space operators. Some of our inequalities improve well-known ones. More precisely, we prove that, if A,B ? B(H) such that A is self-adjoint with ?1 = min ?i ? ?(A) (the spectrum of A) and ?2 = max ?i ? ?(A). Then ?(AB) ?||A||?(B) + (||A|| - |?1 + ?2|/2)DB where DB = inf ??C ||B - ?I||. In particular, if A > 0 and ?(A) ? [k||A||,||A||], then ?(AB) ? (2 - k)||A|| ?(B).

Operator Theory: Advances and Applications, 2005

Journal of the Egyptian Mathematical Society, 2012

Demonstratio Mathematica

In this paper we give a class of finite operators of the form A + K, where A € 2(H) and K is compact. These results are used to generalize the theorem of P.R.Halmos [2, Theorem 7 ] and the result given by J. P. Williams [7, Theorem 5] and we prove that Wo(5a,b) = co<t(6a,b)i where ZUo(5/i,b), co(t(6a,b) denote respectively the numerical range of &a,B an d the convex hull of o(6a b) (the spectrum of Sa,b

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, 2005

2021

Basics A few simple examples J. von-Neumann's theorem and explicit constructions by H. Kosaki Matrices of (un)bounded operators Kato's problem and Mc-Intosh counterexample Naimark-Chernoff counterexample References La technique des matrices d'opérateurs non-bornés pour la construction de contre-exemples Laboratoire d'analyse mathématique et applications : LAMA. Département de mathématiques, Université Oran1 Mercredi 03 mars 2021 M. H. Mortad, Laboratoire d'analyse mathématique et applications : LAMA, Université Oran1 Basics A few simple examples J. von-Neumann's theorem and explicit constructions by H. Kosaki Matrices of (un)bounded operators Kato's problem and Mc-Intosh counterexample Naimark-Chernoff counterexample References Table of contents Basics A few simple examples J. von-Neumann's theorem and explicit constructions by H. Kosaki Matrices of (un)bounded operators Kato's problem and Mc-Intosh counterexample Naimark-Chernoff counterexample References M. H. Mortad, Laboratoire d'analyse mathématique et applications : LAMA, Université Oran1 Basics A few simple examples J. von-Neumann's theorem and explicit constructions by H. Kosaki Matrices of (un)bounded operators Kato's problem and Mc-Intosh counterexample Naimark-Chernoff counterexample References Basics M. H. Mortad, Laboratoire d'analyse mathématique et applications : LAMA, Université Oran1 Basics A few simple examples J. von-Neumann's theorem and explicit constructions by H. Kosaki Matrices of (un)bounded operators Kato's problem and Mc-Intosh counterexample Naimark-Chernoff counterexample References Throughout this talk, H designates a complex Hilbert space. Definition 1 Let A be a linear operator with a domain D(A) (which is a linear subspace of H). We say that A is bounded if ∃α ≥ 0, ∀x ∈ D(A) : Ax ≤ α x. Otherwise, we say that A is unbounded. 1. We say that a linear operator B is an extension of another linear operator A, and we write A ⊂ B, if D(A) ⊂ D(B) and ∀x ∈ D(A) : Ax = Bx. 2. If A and B are two operators with domains D(A) and D(B) respectively, then AB is defined by (AB)x := A(Bx). H. Mortad, Laboratoire d'analyse mathématique et applications : LAMA, Université Oran1 Basics A few simple examples J. von-Neumann's theorem and explicit constructions by H. Kosaki Matrices of (un)bounded operators Kato's problem and Mc-Intosh counterexample Naimark-Chernoff counterexample References for each x in the domain D(AB) = {x ∈ D(B) : Bx ∈ D(A)} = B −1 [D(A)]. Similarly, A + B is defined as (A + B)x := Ax + Bx for x in the domain D(A + B) = D(A) ∩ D(B). 3. Let A be a linear operator with a domain D(A) ⊂ H. We say that A is densely defined if D(A) = H. 4. We say that A is closed if its graph G (A) is closed in H × H. 5. Let A be an injective operator (not necessarily bounded) from D(A) into H. Then A −1 : ran(A) → D(A) is called the inverse of A with domain D(A −1) = ran(A).. H. Mortad, Laboratoire d'analyse mathématique et applications : LAMA, Université Oran1 Basics A few simple examples J. von-Neumann's theorem and explicit constructions by H. Kosaki Matrices of (un)bounded operators Kato's problem and Mc-Intosh counterexample Naimark-Chernoff counterexample References. H. Mortad, Laboratoire d'analyse mathématique et applications : LAMA, Université Oran1 Basics A few simple examples J. von-Neumann's theorem and explicit constructions by H. Kosaki Matrices of (un)bounded operators Kato's problem and Mc-Intosh counterexample Naimark-Chernoff counterexample References When A is densely defined, the previous is equivalent to A ⊂ A *. 9. Say that A is self-adjoint if A = A * , that is, if A is symmetric and D(A) = D(A *). 10. Let A be a symmetric operator with domain D(A) ⊂ H. We say that A is positive if < Ax, x >≥ 0, ∀x ∈ D(A). 11. Let A be a densely defined closed operator. The unique positive square root of A * A is called the absolute value of A and we write |A| = (A * A) 1 2. . H. Mortad, Laboratoire d'analyse mathématique et applications : LAMA, Université Oran1 Basics A few simple examples J. von-Neumann's theorem and explicit constructions by H. Kosaki Matrices of (un)bounded operators Kato's problem and Mc-Intosh counterexample Naimark-Chernoff counterexample References A few simple examples M. H. Mortad, Laboratoire d'analyse mathématique et applications : LAMA, Université Oran1 Basics A few simple examples J. von-Neumann's theorem and explicit constructions by H. Kosaki Matrices of (un)bounded operators Kato's problem and Mc-Intosh counterexample Naimark-Chernoff counterexample References

Journal of Functional Analysis, 2014