FEYNMAN'S OPERATIONAL CALCULI FOR NONCOMMUTING OPERATORS: SPECTRAL THEORY

In recent papers the authors presented the key ideas involved in their approach to Feynman's operational calculi for a system of not necessarily commuting bounded linear operators acting on a Banach space. The central objects of the theory are the disentangling algebra, a commutative Banach algebra, and the disentangling map which carries this commutative structure into the noncommutative algebra of operators. Under assumptions concerning the growth of disentangled exponential expressions, the associated functional calculus for the system of operators is a distribution with compact support which we view as the joint spectrum of the operators with respect to the disentangling map. The spectral properties of the disentangling maps are studied in this paper.

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Common spectral properties of linear operators A and B satisfying AkBkAk = Ak+1 and BkAkBk = Bk+1

Asian-European Journal of Mathematics ◽

10.1142/s1793557119500840 ◽

2019 ◽

Vol 12 (05) ◽

pp. 1950084

Author(s):

Anuradha Gupta ◽

Ankit Kumar

Keyword(s):

Banach Space ◽

Positive Integer ◽

Spectral Properties ◽

Linear Operators ◽

Drazin Inverse ◽

Bounded Linear Operators ◽

Bounded Linear ◽

Cline’S Formula

Let [Formula: see text] and [Formula: see text] be two bounded linear operators on a Banach space [Formula: see text] and [Formula: see text] be a positive integer such that [Formula: see text] and [Formula: see text], then [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] have some common spectral properties. Drazin invertibility and polaroidness of these operators are also discussed. Cline’s formula for Drazin inverse in a ring with identity is also studied under the assumption that [Formula: see text] for some positive integer [Formula: see text].

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Joint spectra of operators on Banach space

Glasgow Mathematical Journal ◽

10.1017/s0017089500006352 ◽

1986 ◽

Vol 28 (1) ◽

pp. 69-72 ◽

Cited By ~ 5

Author(s):

Muneo Chō

Keyword(s):

Banach Space ◽

Point Spectrum ◽

Complex Banach Space ◽

Linear Operators ◽

Bounded Linear Operators ◽

Bounded Linear ◽

Joint Spectrum ◽

Approximate Point Spectrum ◽

Definition Of ◽

Unit Vectors

Let X be a complex Banach space. We denote by B(X) the algebra of all bounded linear operators on X. Let = (T1, …, Tn) be a commuting n-tuple of operators on X. And let στ() and σ″() by Taylor's joint spectrum and the doubly commutant spectrum of , respectively. We refer the reader to Taylor [8] for the definition of στ() and σ″(), A point z = (z1,…, zn) of ℂn is in the joint approximate point spectrum σπ() of if there exists a sequence {xk} of unit vectors in X such that∥(Ti – zi)xk∥→0 as k → ∞ for i = 1, 2,…, n.

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Cline’s formula for g-Drazin inverses in a ring

Filomat ◽

10.2298/fil1908249c ◽

2019 ◽

Vol 33 (8) ◽

pp. 2249-2255

Author(s):

Huanyin Chen ◽

Marjan Abdolyousefi

Keyword(s):

Operator Theory ◽

Spectral Properties ◽

Associative Ring ◽

Linear Operators ◽

Drazin Inverse ◽

Bounded Linear Operators ◽

Bounded Linear ◽

Cline’S Formula

It is well known that for an associative ring R, if ab has g-Drazin inverse then ba has g-Drazin inverse. In this case, (ba)d = b((ab)d)2a. This formula is so-called Cline?s formula for g-Drazin inverse, which plays an elementary role in matrix and operator theory. In this paper, we generalize Cline?s formula to the wider case. In particular, as applications, we obtain new common spectral properties of bounded linear operators.

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Semi-normal operators on uniformly smooth Banach spaces

Glasgow Mathematical Journal ◽

10.1017/s0017089500009356 ◽

1990 ◽

Vol 32 (3) ◽

pp. 273-276 ◽

Cited By ~ 1

Author(s):

Muneo Chō

Keyword(s):

Banach Space ◽

Linear Functional ◽

Dual Space ◽

Complex Banach Space ◽

Linear Operators ◽

Bounded Linear ◽

Uniformly Smooth ◽

Normal Operators ◽

Uniformly Smooth Banach Spaces ◽

The Relationship

In this paper we shall examine the relationship between the numerical ranges and the spectra for semi-normal operators on uniformly smooth spaces.Let X be a complex Banach space. We denote by X* the dual space of X and by B(X) the space of all bounded linear operators on X. A linear functional F on B(X) is called state if ∥F∥ = F(I) = 1. When x ε X with ∥x∥ = 1, we denoteD(x) = {f ε X*:∥f∥ = f(x) = l}.

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Uniqueness of the maximal ideal of operators on the ℓp-sum of ℓ∞n (n ∈ ) for 1 < p < ∞

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004115000766 ◽

2016 ◽

Vol 160 (3) ◽

pp. 413-421 ◽

Cited By ~ 2

Author(s):

TOMASZ KANIA ◽

NIELS JAKOB LAUSTSEN

Keyword(s):

Banach Space ◽

BOUNDED LINEAR OPERATORS ON SPACES IN NORMED DUALITY

Glasgow Mathematical Journal ◽

10.1017/s0017089507003503 ◽

2007 ◽

Vol 49 (1) ◽

pp. 145-154

Author(s):

BRUCE A. BARNES

Keyword(s):

Banach Space ◽

Linear Operator ◽

Bounded Linear Operator ◽

Integral Operators ◽

Continuous Functions ◽

Linear Operators ◽

Bounded Linear Operators ◽

Closed Range ◽

Bounded Linear ◽

Spaces Of Continuous Functions

Abstract.LetTbe a bounded linear operator on a Banach spaceW, assumeWandYare in normed duality, and assume thatThas adjointT†relative toY. In this paper, conditions are given that imply that for all λ≠0, λ−Tand λ −T†maintain important standard operator relationships. For example, under the conditions given, λ −Thas closed range if, and only if, λ −T†has closed range.These general results are shown to apply to certain classes of integral operators acting on spaces of continuous functions.

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On the n-parameter abstract Cauchy problem

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700036169 ◽

2004 ◽

Vol 69 (3) ◽

pp. 383-394

Author(s):

M. Janfada ◽

A. Niknam

Keyword(s):

Banach Space ◽

Cauchy Problem ◽

Unique Solution ◽

Initial Value Problems ◽

Abstract Cauchy Problem ◽

Linear Operators ◽

Uniqueness Of Solution ◽

Initial Value ◽

Bounded Linear ◽

Closed Operators

Let Hi(i = 1, 2, …, n), be closed operators in a Banach space X. The generalised initialvalue problem of the abstract Cauchy problem is studied. We show that the uniqueness of solution u: [0, T1] × [0, T2] × … × [0, Tn] → X of this n-abstract Cauchy problem is closely related to C0-n-parameter semigroups of bounded linear operators on X. Also as another application of C0-n-parameter semigroups, we prove that many n-parameter initial value problems cannot have a unique solution for some initial values.

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Locally compact semi-algebras generated by a commuting operator family

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s016117129400102x ◽

1994 ◽

Vol 17 (4) ◽

pp. 717-724

Author(s):

N. R. Nandakumar ◽

Cornelis V. Vandermee

Keyword(s):

Spectral Properties ◽

Linear Operators ◽

Finite Collection ◽

Operator Family ◽

Locally Compact ◽

Bounded Linear Operators ◽

Bounded Linear ◽

Local Compactness

Conditions are provided for the local compactness of the closed semi-algebra generated by a finite collection of commuting bounded linear operators with equibounded iterates in terms of their joint spectral properties.

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Multiparameter spectral theory and Taylor's joint spectrum in Hilbert space

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500006635 ◽

1988 ◽

Vol 31 (1) ◽

pp. 127-144 ◽

Cited By ~ 4

Author(s):

B. P. Rynne

Keyword(s):

Hilbert Space ◽

Spectral Theory ◽

Linear Operators ◽

Coupled Systems ◽

Bounded Linear Operators ◽

Bounded Linear ◽

Joint Spectrum ◽

Image Position

Let n≧1 be an integer and suppose that for each i= 1,…,n, we have a Hilbert space Hi and a set of bounded linear operators Ti, Vij:Hi→Hi, j=1,…,n. We define the system of operatorswhere λ=(λ1,…,λn)∈ℂn. Coupled systems of the form (1.1) are called multiparameter systems and the spectral theory of such systems has been studied in many recent papers. Most of the literature on multiparameter theory deals with the case where the operators Ti and Vij are self-adjoint (see [14]). The non self-adjoint case, which has received relatively little attention, is discussed in [12] and [13].

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A Note on the Caradus Class of Bounded Linear Operators on a Complex Banach Space

Canadian Journal of Mathematics ◽

10.4153/cjm-1969-066-6 ◽

1969 ◽

Vol 21 ◽

pp. 592-594 ◽

Cited By ~ 4

Author(s):

A. F. Ruston

Keyword(s):

Banach Space ◽

Linear Operator ◽

Finite Number ◽

Bounded Linear Operator ◽

Present Note ◽

Complex Banach Space ◽

Linear Operators ◽

Bounded Linear Operators ◽

Bounded Linear ◽

The Subject

1. In a recent paper (1) on meromorphic operators, Caradus introduced the class of bounded linear operators on a complex Banach space X. A bounded linear operator T is put in the class if and only if its spectrum consists of a finite number of poles of the resolvent of T. Equivalently, T is in if and only if it has a rational resolvent (8, p. 314).Some ten years ago (in May, 1957), I discovered a property of the class g which may be of interest in connection with Caradus' work, and is the subject of the present note.2. THEOREM. Let X be a complex Banach space. If T belongs to the class, and the linear operator S commutes with every bounded linear operator which commutes with T, then there is a polynomial p such that S = p(T).

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