A Note on Quasi-paranormal Operators

Let H H be an infinite dimensional complex Hilbert space, and let B ( H ) \mathcal {B}(H) (resp. C ( H ) \mathcal {C}(H) ) be the algebra of all bounded (resp. compact) linear operators on H H . It is well known that every T ∈ B ( H ) T \in \mathcal {B}(H) has a best approximation from the subspace C ( H ) \mathcal {C}(H) . The purpose of this paper is to study the uniqueness problem concerning the best approximation of a bounded linear operator by compact operators. Our criterion for selecting a unique representative from the set of best approximants is that the representative should commute with T T . In particular, many familiar operators are shown to have zero as a unique commuting best approximant.

The collection of one-dimensional subspaces of an anisotropic Hermitian space is naturally endowed with an orthogonality relation and represents the typical example of what is called an orthogonality space: a set endowed with a symmetric, irreflexive binary relation. We investigate in this paper symmetry properties of orthogonality spaces. We show that two conditions concerning the existence of automorphisms of orthogonality spaces are essentially sufficient to characterise the basic model of quantum physics, the countably infinite dimensional complex Hilbert space.

In this paper we give a different interpretation of Bombieri’s norm. This new point of view allows us to work on a problem posed by Beauzamy about the behavior of the sequence Sn(P ) = supQn [PQn]2, where P is a fixed m−homogeneous polynomial and Qn runs over the unit ball of the Hilbert space of n−homogeneous polynomials. We also study the factor problem for homogeneous polynomials defined on C and we obtain sharp inequalities whenever the number of factors is no greater than N . In particular, we prove that for the product of homogeneous polynomials on infinite dimensional complex Hilbert spaces our inequality is sharp. Finally, we use these ideas to prove that any set {zk}k=1 of unit vectors in a complex Hilbert space for which sup‖z‖=1 |〈z, z1〉 · · · 〈z, zn〉| is minimum must be an orthonormal system.

Let be an infinite dimensional complex Hilbert space and be the algebra of all bounded linear operators on . For , we say T satisfies property (ω) if σa (T) \\ σea (T) = π 00(T), where π 00(T) = {λ ∈ isoσ(T) : 0 < n(T − λI) < ∞}. In this paper, we research on the property (ω) for functions of operators by using the new spectrum σvea (T) which is a variant of the Weyl essential approximate point spectrum. At the same time, the stability of SVEP as well as the relationships between the two parts is also given.

Let H be a separable, infinite dimensional complex Hilbert space, and let B(H) denote the algebra of all bounded linear operators on H. Let K(H) denote the ideal of compact operators on H. For any compact operator A let |A|=(A*A)1,2 and S1(A), s2(A),… be the eigenvalues of |A| in decreasing order and repeatedaccording to multiplicity. If, for some 1&lt;p&lt;∞, si(A)p &lt;∞, we say that A is in the Schatten p-class Cp and ∥A∥p=1/p is the p-norm of A. Hence, C1 is the trace class, C2 is the Hilbert–Schmidt class, and C∞ is the ideal of compact operators K(H).

For bounded linear operators, the study ofWeyl-type theorems and properties has been of significant interest for several non-normal classes of operators. In this paper, we extend this study to a class of unbounded posinormal operators. We define and study the spectral properties of unbounded posinormal and totally posinormal operators defined on an infinite dimensional complex Hilbert space H. For this class, under certain conditions several Weyl-type theorems and related properties are obtained.

Let H be a separable infinite dimensional complex Hilbert space, and let B(H) denote the algebra of all bounded linear operators on H. Let A;B be operators in B(H). In this paper we prove that if A is quasi-class A and B* is invertible quasi-class A and AX = XB, for some X ∈ C2 (the class of Hilbert-Schmidt operators on H), then A*X = XB*. We also prove that if A is a quasi-class A operator and f is an analytic function on a neighborhood of the spectrum of A, then f(A) satisfies generalized Weyl's theorem. Other related results are also given.

Let be an infinite dimensional separable complex Hilbert space and let , where is the Banach algebra of all bounded linear operators on . In this paper we prove the following results. If is a operator, then 1. is a hypercyclic operator if and only if D and for every hyperinvariant subspace of . 2. If is a pure, then is a countably hypercyclic operator if and only if and for every hyperinvariant subspace of . 3. has a bounded set with dense orbit if and only if for every hyperinvariant subspace of , .

Let H be an infinite dimensional complex Hilbert space and let be a surjective linear map on B(H) with (I) I 2 K(H), where K(H) denotes the closed ideal of all compact operators on H. If preserves the set of upper semi-Weyl operators and the set of all normal eigenvalues in both directions, then is an automorphism of the algebra B(H). Also the relation between the linear maps preserving the set of upper semi-Weyl operators and the linear maps preserving the set of left invertible operators is considered.

Let A and B be quasisimilar seminormal operators on a separable, infinite dimensional complex Hilbert space. Several conditions which imply that A and B have equal essential spectra are presented. For example, if A and B are both biquasitriangular then A and B have equal essential spectra.

Let 𝒜 {\mathcal{A}} be a standard operator algebra containing the identity operator I on an infinite dimensional complex Hilbert space ℋ {\mathcal{H}} which is closed under adjoint operation. Suppose that ϕ : 𝒜 → 𝒜 {\phi:\mathcal{A}\to\mathcal{A}} is the second nonlinear mixed Lie triple derivation. Then ϕ is an additive ∗ {\ast} -derivation.

The goal of this paper is to confirm that the unitary group U(H) on an infinite dimensional complex Hilbert space is a topological group in its strong topology, and to emphasize the importance of this property for applications in topology. In addition, it is shown that U(H) in its strong topology is metrizable and contractible if H is separable. As an application Hilbert bundles are classified by homotopy.

Let H be a complex infinite dimensional Hilbert space. For T ? B(H), T is said to satisfy property (UW?) if the complement in the approximate point spectrum of the Weyl essential approximate point spectrum coincides with the poles of T. In this paper, we deeply talk about the property (UW?) under some perturbations and property (UW?) for functions of operators. In addition, if T is Drazin invertible, then property (UW?) for functions of T can be transmitted to the functions of its Drazin inverse.

Finite dimensional approximation and Newton-based algorithm for stochastic approximation in Hilbert space

Sequences of [formula omitted]-term deviations in Hilbert space