scholarly journals Analyticity of the Resolvent Function

2021 ◽  
pp. 1-9
Author(s):  
Achiles Nyongesa Simiyu ◽  
Philis Alosa ◽  
Fanuel Olege
Keyword(s):  
Linear Operator ◽  
Bounded Linear Operator ◽  
Complex Banach Space ◽  
Analytic Dependence ◽  
Bounded Linear ◽  
Resolvent Set ◽  
Early Signal ◽  
Definition Of ◽  
Resolvent Function ◽  
Differential And Integral Equations

Analytic dependence on a complex parameter appears at many places in the study of differential and integral equations. The display of analyticity in the solution of the Fredholm equation of the second kind is an early signal of the important role which analyticity was destined to play in spectral theory. The definition of the resolvent set is very explicit, this makes it seem plausible that the resolvent is a well behaved function. Let T be a closed linear operator in a complex Banach space X. In this paper we show that the resolvent set of T is an open subset of the complex plane and the resolvent function of T is analytic. Moreover, we show that if T is a bounded linear operator, the resolvent function of T is analytic at infinity, its value at infinity being 0 (where 0 is the bounded linear operator 0 in X). Consequently, we also show that if T is bounded in X then the spectrum of T is non-void.

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

1969 ◽  
Vol 21 ◽  
pp. 592-594 ◽  
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).

scholarly journals Expansion of analytic functions of an operator in series of Faber polynomials

1997 ◽  
Vol 56 (2) ◽  
pp. 303-318 ◽  
Author(s):  
Maurice Hasson
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Analytic Functions ◽  
Bounded Linear Operator ◽  
Complex Banach Space ◽  
Faber Polynomials ◽  
Bounded Linear ◽  
In Series ◽  
Image Position

Let T: B → B be a bounded linear operator on the complex Banach space B and let f(z) be analytic on a domain D containing the spectrum Sp(T) of T. Then f(T) is defined bywhere C is a contour surrounding SP(T) and contained in D.

scholarly journals Perturbations of local C-cosine functions

2020 ◽  
Vol 65 (4) ◽  
pp. 585-597
Author(s):  
Chung-Cheng Kuo
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Complex Banach Space ◽  
Cosine Function ◽  
Bounded Linear ◽  
Cosine Functions

"We show that $\tA+\tB$ is a closed subgenerator of a local $\tC$-cosine function $\tT(\cdot)$ on a complex Banach space $\tX$ defined by $$\tT(t)x=\sum\limits_{n=0}^\infty \tB^n\int_0^tj_{n-1}(s)j_n(t-s)\tC(|t-2s|)xds$$ for all $x\in\tX$ and $0\leq t<T_0$, if $\tA$ is a closed subgenerator of a local $\tC$-cosine function $\tC(\cdot)$ on $\tX$ and one of the following cases holds: $(i)$ $\tC(\cdot)$ is exponentially bounded, and $\tB$ is a bounded linear operator on $\overline{\tD(\tA)}$ so that $\tB\tC=\tC\tB$ on $\overline{\tD(\tA)}$ and $\tB\tA\subset\tA\tB$; $(ii)$ $\tB$ is a bounded linear operator on $\overline{\tD(\tA)}$ which commutes with $\tC(\cdot)$ on $\overline{\tD(\tA)}$ and $\tB\tA\subset\tA\tB$; $(iii)$ $\tB$ is a bounded linear operator on $\tX$ which commutes with $\tC(\cdot)$ on $\tX$. Here $j_n(t)=\frac{t^n}{n!}$ for all $t\in\Bbb R$, and $$\int_0^tj_{-1}(s)j_0(t-s)\tC(|t-2s|)xds=\tC(t)x$$ for all $x\in\tX$ and $0\leq t<T_0$."

scholarly journals Non-recurrence sets for weakly mixing linear dynamical systems

2012 ◽  
Vol 34 (1) ◽  
pp. 132-152 ◽  
Author(s):  
SOPHIE GRIVAUX
Keyword(s):  
Banach Space ◽  
Dynamical Systems ◽  
Linear Operator ◽  
Gaussian Measure ◽  
Bounded Linear Operator ◽  
Probability Space ◽  
Complex Banach Space ◽  
Bounded Linear ◽  
Linear Dynamical Systems ◽  
Weakly Mixing

AbstractWe study non-recurrence sets for weakly mixing dynamical systems by using linear dynamical systems. These are systems consisting of a bounded linear operator acting on a separable complex Banach space$X$, which becomes a probability space when endowed with a non-degenerate Gaussian measure. We generalize some recent results of Bergelson, del Junco, Lemańczyk and Rosenblatt, and show in particular that sets$\{n_{k}\}$such that$n_{k+1}/n_{k}\to +\infty $, or such that$n_{k}$divides$n_{k+1}$for each$k\ge 0$, are non-recurrence sets for weakly mixing linear dynamical systems. We also give examples, for each$r\ge 1$, of$r$-Bohr sets which are non-recurrence sets for some weakly mixing systems.

scholarly journals On generalization of decomposability

1981 ◽  
Vol 22 (1) ◽  
pp. 77-81 ◽  
Author(s):  
Ridgley Lange
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Invariant Subspace ◽  
Bounded Linear Operator ◽  
Invariant Subspaces ◽  
Complex Banach Space ◽  
Open Cover ◽  
Bounded Linear ◽  
Image Position ◽  
Finite Open Cover

Let X be a complex Banach space and let T be a bounded linear operator on X. Then T is decomposable if for every finite open cover of σ(T) there are invariant subspaces Yi(i= 1, 2, …, n) such that(An invariant subspace Y is spectral maximal [for T] if it contains every invariant subspace Z for which σ(T|Z) ⊂ σ(T|Y).).

On Certain Classes of Bounded Linear Operators

1970 ◽  
Vol 13 (4) ◽  
pp. 469-473
Author(s):  
C-S Lin
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Banach Algebra ◽  
Fredholm Operator ◽  
Bounded Linear Operator ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Complex Numbers ◽  
Bounded Linear Operators ◽  
Bounded Linear

Let T—c be a Fredholm operator, where T is a bounded linear operator on a complex Banach space and c is a scalar, the set of all such scalars is called the Φ-set of T [2] and was studied by many authors. In this connection, the purpose of the present paper is to investigate some classes Φ(V) of all such operators for any subset V of the complex plane.Let X be a Banach space over the field C of complex numbers with dim Z = ∞, unless otherwise stated, B(X) the Banach algebra of all bounded linear operators and K(X) the closed two-sided ideal of all compact operators on X.

scholarly journals A result on hermitian operators

1989 ◽  
Vol 31 (1) ◽  
pp. 71-72
Author(s):  
J. E. Jamison ◽  
Pei-Kee Lin
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Numerical Range ◽  
Complex Banach Space ◽  
Bounded Linear ◽  
Image Position

Let X be a complex Banach space. For any bounded linear operator T on X, the (spatial) numerical range of T is denned as the setIf V(T) ⊆ R, then T is called hermitian. Vidav and Palmer (see Theorem 6 of [3, p. 78] proved that if the set {H + iK:H and K are hermitian} contains all operators, then X is a Hilbert space. It is natural to ask the following question.

scholarly journals The n-inverses of a matrix

Filomat ◽  
2017 ◽  
Vol 31 (12) ◽  
pp. 3801-3813
Author(s):  
Caixing Gu ◽  
Heidi Keas ◽  
Robert Lee
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Tensor Products ◽  
Complex Banach Space ◽  
Spectral Characterization ◽  
Bounded Linear ◽  
Nilpotent Operators ◽  
Nilpotent Matrices

The concept of a left n-inverse of a bounded linear operator on a complex Banach space was introduced recently. Previously, there have been results on products and tensor products of left n-inverses, and the representation of left n-inverses as the sum of left inverses and nilpotent operators was being discussed. In this paper, we give a spectral characterization of the left n-inverses of a finite (square) matrix. We also show that a left n-inverse of a matrix T is the sum of the inverse of T and two nilpotent matrices.

scholarly journals Perturbation Theory for Property (VE) and Tensor Product

Mathematics ◽  
2021 ◽  
Vol 9 (21) ◽  
pp. 2775
Author(s):  
Elvis Aponte ◽  
José Sanabria ◽  
Luis Vásquez
Keyword(s):  
Banach Space ◽  
Perturbation Theory ◽  
Linear Operator ◽  
Tensor Product ◽  
Bounded Linear Operator ◽  
Sufficient Conditions ◽  
Complex Banach Space ◽  
Bounded Linear

Given a complex Banach space X, we investigate the stable character of the property (VE) for a bounded linear operator T:X→X, under commuting perturbations that are Riesz, compact, algebraic and hereditarily polaroid. We also analyze sufficient conditions that allow the transfer of property (VE) from the tensorial factors T and S to its tensor product.

scholarly journals A direct proof of a theorem of West on sequences of Riesz operators

1974 ◽  
Vol 15 (2) ◽  
pp. 93-94
Author(s):  
Anthony F. Ruston
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Direct Proof ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Riesz Operator ◽  
Bounded Linear ◽  
Riesz Operators

We recall (cf. [2] Definitions 3.1 and 3.2, p. 322) that a bounded linear operator T on a Banach space ℵ into itself is said to be asymptotically quasi-compact if K(Tn)⅟n → 0 as n → ∞. where K(U) = inf ∥U–C∥ for every bounded linear operator U on ℵ into itself, the infimum being taken over all compact linear operators C on ℵ into itself. For a complex Banach space, this is equivalent (cf. [2], pp. 319, 321 and 326) to T being a Riesz operator.

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