A note on property (UWΠ)

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.

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.

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.

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 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 T be a bounded linear operator on an infinite dimensional complex Hilbert space. In this paper, we introduce the new class, denoted \({{\mathcal{QP}}}\) , of operators satisfying \({{\|T^{2}x\|^{2}\leq \|T^{3}x\|\|Tx\|}}\) for all \({{x \in \mathcal{H}}}\). This class includes the classes of paranormal operators and quasi-class A operators. We prove basic structural properties of these operators. Using these results, we also prove that if E is the Riesz idempotent for a nonzero isolated point λ0 of the spectrum of \({{T \in \mathcal{QP}}}\) , then E is self-adjoint if and only if \({{N(T-\lambda_{0}) \subseteq N(T^{*}-\overline{\lambda}_{0})}}\).

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).

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.

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 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 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.

Introduction. It has been known for some time that on a complex infinite-dimensional Hilbert space there exist invertible operators without square roots, indeed without roots of any order, which therefore do not belong to the range of the exponential function. A first class of examples of such operators was described by Halmos, Lumer, and Schaffer [3]: the space considered was the separable Hilbert space of all complex-valued functions defined, analytic, and squaresummable on a domain D of the complex plane (with the L2-norm); the operator was the analytic position operator A defined by (A+5) (z) =zcf(z), zCD. It was shown that A is invertible and lacks a square root (indeed, a root of any order) if and only if D surrounds the origin but does not contain it. Halmos and Lumer [2] used the concept of multiplicity to show that the analytic position operator for such D is an interior point (in the norm topology for operators) of the set of invertible operators without roots. Recently, Deckard and Pearcy [1] described another class of invertible square-root-less operators on a separable Hilbert space; the spectral properties of these operators are quite different from those of the operators discussed in [3] and [2]. We shall not be concerned with this class of examples in this paper. Root-less operators are beginning to be used in various contexts: we mention, in particular, the disproof by Massera and Schaffer [5, p. 92], of an extension to Hilbert space of Floquet's Theorem on periodic differential equations. It appears that more information about such operators and more flexibility in their choice is desirable. To take the analytic position operator, for instance: there is of course some latitude in the choice of the domain D, but the description of the operator in terms of an orthonormal basis is awkward even in the simplest case, when D is an annulus centred at the origin, and quite unmanageable in other cases; besides, this description can not be perturbed or altered in any prescribed way as the application in hand might require. We remark, however, that the original definition has the advantage of making almost obvious the spectral structure that ensures the lack of roots.

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 , .

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