scholarly journals Well-bounded operators of type (B) in a class of Banach spaces

1987 ◽  
Vol 42 (3) ◽  
pp. 399-408 ◽  
Author(s):  
Werner Ricker
Keyword(s):  
Integral Representation ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Unitary Groups ◽  
Real Spectrum ◽  
Type B ◽  
Bounded Operators ◽  
Scalar Type ◽  
Normal Operators ◽  
Grothendieck Space

AbstractIt is shown that in a Grothendieck space with the Dunford-Pettis property, the class of well-bounded operators of type (B) coincides with the class of scalar-type spectral operators with real spectrum. It turns out that in such Banach spaces, analogues of the classical theorems of Hille-Sz. Nagy and Stone concerned with the integral representation of C0-semigroups of normal operators and strongly continuous unitary groups in Hilbert spaces, respectively, are of a very special nature.

scholarly journals Commuting well-bounded operators on Hilbert spaces

1976 ◽  
Vol 20 (2) ◽  
pp. 167-172 ◽  
Author(s):  
T. A. Gillespie
Keyword(s):  
Integral Representation ◽  
Hilbert Spaces ◽  
Reflexive Banach Space ◽  
Adjoint Operator ◽  
Complex Polynomial ◽  
Real Spectrum ◽  
Bounded Operators ◽  
Bounded Linear ◽  
Scalar Type ◽  
Image Position

A bounded linear operator T on a complex reflexive Banach space is said to be well-bounded if it is possible to choose a compact interval J = [a, b] and a positive constant M such thatfor every complex polynomial p, where ‖p‖J denotes sup {|p(t)|:t ∈ J}. Such operators were introduced and first studied by Smart (4). They are of interest principally because they admit (and in fact are characterised by) an integral representation similar to, but in general weaker than, the integral representation of a self-adjoint operator on a Hilbert space. (See (2) and (4) for details.) It is easily seen, by verifying (1) directly, that T is well-bounded if it is a scalar-type spectral operator with real spectrum.

scholarly journals Joint spectra of commuting normal operators on Banach spaces

1988 ◽  
Vol 30 (3) ◽  
pp. 339-345 ◽  
Author(s):  
Muneo Chō
Keyword(s):  
Functional Analysis ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Functional Calculus ◽  
Joint Spectrum ◽  
Normal Operators

The joint spectrum for a commuting n-tuple in functional analysis has its origin in functional calculus which appeared in J. L. Taylor's epoch-making paper [19] in 1970. Since then, many papers have been published on commuting n-tuples of operators on Hilbert spaces (for example, [3], [4], [5], [8], [9], [10], [21], [22]).

Normal operators on Banach spaces

1979 ◽  
Vol 20 (2) ◽  
pp. 163-168 ◽  
Author(s):  
Che-Kao Fong
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Bounded Linear Operator ◽  
Bounded Linear ◽  
Normal Operators

A (bounded, linear) operator H on a Banach space is said to be hermitian if ∥exp(itH)∥ = 1 for all real t. An operator N on is said to be normal if N = H + iK, where H and K are commuting hermitian operators. These definitions generalize those familiar concepts of operators on Hilbert spaces. Also, the normal derivations defined in [1] are normal operators. For more details about hermitian operators and normal operators on general Banach spaces, see [4]. The main result concerning normal operators in the present paper is the following theorem.

C-spectral operators

1974 ◽  
Vol 76 (1) ◽  
pp. 67-113
Author(s):  
K. Dayanithy
Keyword(s):  
Functional Analysis ◽  
Banach Spaces ◽  
Spectral Theory ◽  
Canonical Form ◽  
Hilbert Spaces ◽  
Normal Operators ◽  
Considerable Work ◽  
Almost All ◽  
Spectral Theory Of Operators

The importance of the spectral theory of operators in Functional Analysis cannot be over emphasized. Spectral theory is in its best form when one considers normal operators in Hilbert spaces. However, for te dimensional spaces one has the reduction to Jordan's Canonical form. In an attempt to generalize this reduction for arbitrary Banach spaces, Dunford introduced the concept of spectral operators. Considerable work has been done in recent times in the study of spectral theory in Banach spaces, almost all of which stems from the pioneering work of Dunford.

scholarly journals Invertible Operators on Certain Banach Spaces

1977 ◽  
Vol 20 (2) ◽  
pp. 153-160
Author(s):  
J.-M. Belley
Keyword(s):  
Banach Spaces ◽  
Hilbert Spaces ◽  
Unitary Operator ◽  
Invertible Operator ◽  
Fourier Series Expansion ◽  
Unitary Operators ◽  
Bounded Operators ◽  
Power Bounded Operators ◽  
Resolution Of The Identity ◽  
Power Bounded

It has long been the practice in the theory of Hilbert spaces to use the Fourier series expansion (i.e. the Levy inversion formula) for the resolution of the identity associated with a unitary operator to obtain results for this operator, and hence for any power bounded invertible operator on such spaces since they are necessarily isomorphic to unitary operators [5, p. 1945]. Though many important power bounded operators on Banach spaces are not spectral [6, p. 1045-1051] the approach of this paper permits us to deduce for such operators results similar to those known for spectral operators.

scholarly journals Sets of periods for chaotic linear operators

2021 ◽  
Vol 115 (2) ◽  
Author(s):  
J. A. Conejero ◽  
F. Martínez-Giménez ◽  
A. Peris ◽  
F. Rodenas
Keyword(s):  
Banach Spaces ◽  
Hilbert Spaces ◽  
Complete Characterization ◽  
Linear Operators

AbstractWe provide a complete characterization of the possible sets of periods for Devaney chaotic linear operators on Hilbert spaces. As a consequence, we also derive this characterization for linearizable maps on Banach spaces.

scholarly journals The dual of the space of bounded operators on a Banach space

2021 ◽  
Vol 8 (1) ◽  
pp. 48-59
Author(s):  
Fernanda Botelho ◽  
Richard J. Fleming
Keyword(s):  
Banach Space ◽  
Tensor Product ◽  
Banach Spaces ◽  
Dual Space ◽  
Tensor Products ◽  
Bounded Operators ◽  
Projective Tensor Product ◽  
Projective Tensor

Abstract Given Banach spaces X and Y, we ask about the dual space of the 𝒧(X, Y). This paper surveys results on tensor products of Banach spaces with the main objective of describing the dual of spaces of bounded operators. In several cases and under a variety of assumptions on X and Y, the answer can best be given as the projective tensor product of X ** and Y *.

scholarly journals The spectral theorem for well-bounded operators

1993 ◽  
Vol 54 (3) ◽  
pp. 334-351 ◽  
Author(s):  
Ian Doust ◽  
Qiu Bozhou
Keyword(s):  
Functional Calculus ◽  
Continuous Functions ◽  
Weak Compactness ◽  
Compact Interval ◽  
Spectral Theorem ◽  
Type B ◽  
Bounded Operators ◽  
Absolutely Continuous Functions ◽  
Absolutely Continuous

AbstractWell-bounded operators are those which possess a bounded functional calculus for the absolutely continuous functions on some compact interval. Depending on the weak compactness of this functional calculus, one obtains one of two types of spectral theorem for these operators. A method is given which enables one to obtain both spectral theorems by simply changing the topology used. Even for the case of well-bounded operators of type (B), the proof given is more elementary than that previously in the literature.

scholarly journals A characterisation of Hilbert spaces via orthogonality and proximinality

2005 ◽  
Vol 71 (1) ◽  
pp. 107-111
Author(s):  
Fathi B. Saidi
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Real Banach Space ◽  
Nonzero Element ◽  
Dimensional Subspace ◽  
Two Dimensional ◽  
Orthogonal Direction ◽  
Open Problems

In this paper we adopt the notion of orthogonality in Banach spaces introduced by the author in [6]. There, the author showed that in any two-dimensional subspace F of E, every nonzero element admits at most one orthogonal direction. The problem of existence of such orthogonal direction was not addressed before. Our main purpose in this paper is the investigation of this problem in the case where E is a real Banach space. As a result we obtain a characterisation of Hilbert spaces stating that, if in every two-dimensional subspace F of E every nonzero element admits an orthogonal direction, then E is isometric to a Hilbert space. We conclude by presenting some open problems.

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