On typical properties of Hilbert space operators

2014

Denote by [0, ω1) the locally compact Hausdorff space consisting of all countable ordinals, equipped with the order topology, and let C0[0, ω1) be the Banach space of scalar-valued, continuous functions which are defined on [0, ω1) and vanish eventually. We show that a weak *compact subset of the dual space of C0[0, ω1) is either uniformly Eberlein compact, or it contains a homeomorphic copy of a particular form of the ordinal interval [0, ω1]. This dichotomy yields a unifying approach to most of the existing studies of the Banach space C0[0, ω1) and the Banach algebra B(C0[0, ω1)) of bounded, linear operators acting on it, and it leads to several new results, as well as to stronger versions of known ones. Specifically, we deduce that a Banach space which is a quotient of C0[0, ω1) can either be embedded in a Hilbert-generated Banach space, or it is isomorphic to the direct sum of C0[0, ω1) and a subspace of a Hilbert-generated Banach space; and we obtain several equivalent conditions describing the Loy-Willis ideal M , which is the unique maximal ideal of B(C0[0, ω1)), including the following: an operator belongs to M if and only if it factors through the Banach space ( α<ω 1 C[0, α])c 0 . Among the consequences of these characterizations of M is that M has a bounded left approximate identity; this resolves a problem left open by Loy and Willis.

Journal of the Australian Mathematical Society, 2016

We establish a spectral characterization theorem for the operators on complex Hilbert spaces of arbitrary dimensions that attain their norm on every closed subspace. The class of these operators is not closed under addition. Nevertheless, we prove that the intersection of these operators with the positive operators forms a proper cone in the real Banach space of hermitian operators.

It is known that on a Hilbert space, the sum of a real scalar-type operator and a commuting well-bounded operator is well-bounded. The corresponding property has been shown to be fail on L p spaces, for 1 < p = 2 < ∞. We show that it does hold however on every Banach space X such that X or X * is a Grothendieck space. This class notably includes L 1 and C(K) spaces. MSC ( ): 47B40, 46B03, 46B20 (primary) 46B10, 46B22, 46B26 (secondary).

arXiv: Functional Analysis, 2014

Over the decades, Functional Analysis has been enriched and inspired on account of demands from neighboring fields, within mathematics, harmonic analysis (wavelets and signal processing), numerical analysis (finite element methods, discretization), PDEs (diffusion equations, scattering theory), representation theory; iterated function systems (fractals, Julia sets, chaotic dynamical systems), ergodic theory, operator algebras, and many more. And neighboring areas, probability/statistics (for example stochastic processes, Ito and Malliavin calculus), physics (representation of Lie groups, quantum field theory), and spectral theory for Schr\&quot;odinger operators. We have strived for a more accessible book, and yet aimed squarely at applications; -- we have been serious about motivation: Rather than beginning with the four big theorems in Functional Analysis, our point of departure is an initial choice of topics from applications. And we have aimed for flexibility of use; acknowledgi...

Journal of the London Mathematical Society, 1989

Several classes of operators have been suggested as generalisations of self-adjoint operators to the Banach space setting. The two classes which we shall be concerned with here, well-bounded and scalar-type spectral operators, both give structure theorems similar to the spectral theorem for self-adjoint operators. The main aim of this paper is to examine the relationship between these two classes on L v spaces. Fong and Lam have proved the following theorem [10, Proposition 2.15]. THEOREM 1.1. Suppose that T is a bounded linear operator on a Hilbert space, and that there exists a compact interval [a, b] a U. such that for all polynomials g \\g(T)\\^\g(a)\+{ b \g'(t)\dt.

Journal of Mathematical Analysis and Applications, 2007

We find necessary and sufficient conditions for a Banach space operator T to satisfy the generalized Browder's theorem, and we obtain new necessary and sufficient conditions to guarantee that the spectral mapping theorem holds for the B-Weyl spectrum and for polynomials in T . We also prove that the spectral mapping theorem holds for the B-Browder spectrum and for analytic functions on an open neighborhood of σ(T ). As applications, we show that if T is algebraically M -hyponormal, or if T is algebraically paranormal, then the generalized Weyl's theorem holds for f (T ), where f ∈ H((T )), the space of functions analytic on an open neighborhood of σ(T ). We also show that if T is reduced by each of its eigenspaces, then the generalized Browder's theorem holds for f (T ), for each f ∈ H(σ(T )).

Turkish Journal of Mathematics, 2023

We introduce the class of (M, k) -quasi- * -paranormal operators on a Hilbert space H . This class extends the classes of * -paranormal and k -quasi- * -paranormal operators. An operator T on a complex Hilbert space is called for all x ∈ H. In the present article, we give operator matrix representation of a (M, k) -quasi- * -paranormal operator. The compactness, the invariant subspace, and some topological properties of this class of operators are studied. Several properties of this class of operators are also presented.

Mediterranean Journal of Mathematics

In this article, we introduce the notion of polynomial demicompactness and we use it to give some results on Fredholm operators and to establish a fine description of some essential spectra of a closed densely defined linear operator. Our work is a generalization of many known ones in the literature.

Linear and Multilinear Algebra, 2019

In this study, some problems of operator theory on the reproducing kernel Hilbert space by using the Berezin symbols method are investigated. Namely, invariant subspaces of weighted composition operators on H 2 are studied. Moreover, some new inequalities for the Berezin number of operators are proved. In particular, new reverse inequalities for the Berezin numbers ber |A| 2 and ber (A) of operators |A| 2 = A * A and A on the reproducing kernel Hilbert space are given. Also, reverse inequalities for the Berezin number of two operators are proved. Under some conditions we prove the power inequality ber A n ≤ (ber (A)) n which is related to a well known analogue estimate of Halmos for the numerical radius.

Proceedings of the American Mathematical Society, 1972

If H is the Hilbert transform on LP(Z), then T=ttI+íH is a well-bounded operator for \<p<ao, but is not a scalar-type spectral operator except when/>=2. The purpose of this note is to show that there is a well-bounded operator on a reflexive Banach space which is not scalar-type spectral.

2020

This present paper deals with the study of Hilbert Space and Algebra of Operators. Here, we consider R as additive group of reals with discrete topology and several ways of constructing C* algebras Canonically associated with R and π, The Universal representation of R on Hilbert Space H, it is proved in this paper that all C*algebras homomorphism and representation will be * preserving

2020

In this paper we investigate results on unitary equivalence of operators that include n-binormal, skew binormal and n-power-hyponormal operators acting on complex Hilbert space H. AMS subject classification 47B47, 47A30, 47B20.

2016

Journal of Logic and Analysis, 2011

p-Adic Numbers, Ultrametric Analysis and Applications, 2019

This work will be centered in commutative Banach subalgebras of the algebra of bounded linear operators defined on free Banach spaces of countable type. The main goal of this work will be to formulate a representation theorem for these operators through integrals defined by spectral measures type. In order to get this objective, we will show that, under special conditions, each one of these algebras is isometrically isomorphic to some space of continuous functions defined over a compact set. Then, we will identify such compact sets developing the Gelfand space theory in the non-Archimedean setting. This fact will allow us to define a measure which is known as spectral measure. As a second goal, we will formulate a matrix representation theorem for this class of operators in which the entries of the matrices will be integrals coming from scalar measures.