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Journal ArticleDOI

Level sets of the condition spectrum

01 Aug 2017-Annals of Functional Analysis (Tusi Mathematical Research Group)-Vol. 8, Iss: 3, pp 314-328
TL;DR: For 0 < e ≤ 1 and an element a of a complex unital Banach algebra A, the authors showed that the e-level set of the condition spectrum has an empty interior in the unbounded component of the resolvent set of a.
Abstract: For 0 < e ≤ 1 and an element a of a complex unital Banach algebra A, we prove the following two topological properties about the level sets of the condition spectrum. (1) If e = 1, then the 1-level set of the condition spectrum of a has an empty interior unless a is a scalar multiple of the unity. (2) If 0 < e < 1, then the e-level set of the condition spectrum of a has an empty interior in the unbounded component of the resolvent set of a. Further, we show that, if the Banach space X is complex uniformly convex or if X* is complex uniformly convex, then, for any operator T acting on X, the level set of the e-condition spectrum of T has an empty interior.

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Citations
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Journal ArticleDOI
01 Jun 2021
TL;DR: In this paper, the authors discuss the properties of the spectrum of a complex unital Banach algebra and the effects of perturbations on these sets and also discuss the question of determining the property of the element a from the knowledge of these sets.
Abstract: In order to understand the behaviour of a square matrix or a bounded linear operator on a Banach space or more generally an element of a Banach algebra, some subsets of the complex plane are associated with such an object. Most popular among these sets is the spectrum $$\sigma (a) $$ of an element a in a complex unital Banach algebra A with unit 1 defined as follows: $$\begin{aligned} \sigma (a) := \{\lambda \in \mathbb {C}: \lambda - a \,\, \text{ is } \text{ not } \text{ invertible } \text{ in }\, A\}. \end{aligned}$$ Here and also in what follows, we identify $$\lambda .1$$ with $$\lambda $$ . Also quite popular is Numerical range V(a) of a. This is defined as follows: $$\begin{aligned} V(a) := \{ \phi (a): \phi \,\, \text{ is } \text{ a } \text{ continuous } \text{ linear } \text{ functional } \text{ on } \,\, A \, \, \text{ satisfying } \, \, \Vert \phi \Vert = 1 = \phi (1) \}. \end{aligned}$$ Then there are many generalizations, modifications, approximations etc. of the spectrum. Let $$\epsilon > 0$$ and n a nonnegative integer. These include $$\epsilon -$$ condition spectrum $$\sigma _{\epsilon }(a)$$ , $$\epsilon -$$ pseudospectrum $$\Lambda _{\epsilon }(a)$$ and $$(n, \epsilon )-$$ pseudospectrum $$\Lambda _{n, \epsilon }(a)$$ . These are defined as follows: $$\begin{aligned} \sigma _{\epsilon }(a) := \left\{ \lambda \in \mathbb {C}: \Vert \lambda - a\Vert \Vert (\lambda -a)^{-1}\Vert \ge \frac{1}{\epsilon } \right\} \end{aligned}$$ In this and the following definitions we follow the convention : $$ \Vert (\lambda -a)^{-1}\Vert = \infty $$ if $$\lambda - a$$ is not invertible. $$\begin{aligned}&\Lambda _{\epsilon }(a) := \left\{ \lambda \in \mathbb {C}: \Vert (\lambda -a)^{-1}\Vert \ge \frac{1}{\epsilon } \right\} \\&\Lambda _{n, \epsilon }(a) := \left\{ \lambda \in \mathbb {C}: \Vert (\lambda -a)^{-2^n}\Vert ^{ 1/2^n } \ge \frac{1}{\epsilon } \right\}. \end{aligned}$$ In this survey article, we shall review some basic properties of these sets, relations among these sets and also discuss the effects of perturbations on these sets and the question of determining the properties of the element a from the knowledge of these sets.

2 citations

Journal ArticleDOI
TL;DR: In this article, the upper and lower hemicontinuity and joint continuity of the condition spectrum and its level set maps in appropriate settings were studied, and it was shown that the empty interior of a condition spectrum at a given point in a complex unital Banach algebra plays a pivotal role in the continuity of required maps at that point.
Abstract: Let ${\\mathcal{A}}$ be a complex unital Banach algebra, let $a$ be an element in it and let $0<\\unicode[STIX]{x1D716}<1$. In this article, we study the upper and lower hemicontinuity and joint continuity of the condition spectrum and its level set maps in appropriate settings. We emphasize that the empty interior of the $\\unicode[STIX]{x1D716}$-level set of a condition spectrum at a given $(\\unicode[STIX]{x1D716},a)$ plays a pivotal role in the continuity of the required maps at that point. Further, uniform continuity of the condition spectrum map is obtained in the domain of normal matrices.

2 citations


Cites background from "Level sets of the condition spectru..."

  • ...The central theme of [13] is to identify and classify the Banach algebras A in which the interior of L (a) is empty for any a ∈ A and ∈ (0, 1)....

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  • ...Further, for ∈ (0, 1], the -level set of the condition spectrum of a ∈ A is defined in [13] as the following set....

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Dissertation
01 Jan 2018
TL;DR: In this article, the Banach algebras at which the interior of the level set of the condition spectrum is empty are identified and the continuity results of the spectrum are obtained.
Abstract: For 0 < � < 1 and a Banach algebra element a, this thesis aims to establish the results related to continuity of condition spectrum and its level set correspondence at (�; a). Here we propose a method of study to achieve the continuity. We first identify the Banach algebras at which the interior of the level set of condition spectrum is empty and then we obtain the continuity results. This thesis consists of four chapters. Chapter 1 contains all the prerequisites which are crucial for the development of the thesis. In particular, this chapter has a quick review of the basic properties of spectrum, condition spectrum, upper and lower hemicontiuous correspondences. We also concentrate on analytic vector valued maps and generalized maximum modulus theorem for them. For an element a in A, Chapter 2 has the results related to interior of the level of set of the condition spectrum of a. At first, we focus on 1

1 citations


Additional excerpts

  • ...3 in [45], interior of L 0(A) is empty in the set {λ ∈ C : |λ| > 1}....

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References
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Book ChapterDOI

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01 Jan 2012

139,059 citations


Additional excerpts

  • ...If there exists a ∈ A \ Ce such that σ(a) = {λ1, λ2} with λ1 6= λ2, then by proposition 9 in §7 of [8], there exists idempotents e1 and e2 such that σ (ae1) = {λ1}, σ (ae2) = {λ2} and a = ae1 + ae2....

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Journal ArticleDOI

850 citations


"Level sets of the condition spectru..." refers background in this paper

  • ...It is proved in [5] that Hilbert spaces and Lp (with 1 < p < ∞) spaces are uniformly convex Banach spaces and hence they are all complex uniformly convex Banach spaces....

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Journal ArticleDOI

105 citations


"Level sets of the condition spectru..." refers background in this paper

  • ...One can find, some more answers related to this question in [2], [3] and [4]....

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Book
26 Oct 1999
TL;DR: In this article, the spectral theory of differential operators is studied and a general spectral theory for general differential operators for general operators is presented, including generalized eigenfunctions and spectral projections of a differential operator.
Abstract: Unbounded linear operators Fredholm operators Introduction to the spectral theory of differential operators Principal part of a differential operator Projections and generalized eigenfunction expansions Spectral theory for general differential operators Bibliography Index.

99 citations


Additional excerpts

  • ...A function f : Ω→ A is said to be differentiable at the point μ ∈ Ω (see [13], Definition 3....

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Journal ArticleDOI
01 Jan 1975
TL;DR: Theorem 1. as mentioned in this paper shows that the complex space L.(S, a, p) is uniformly c-convex, which is a generalization of strict and uniform convexity.
Abstract: Strict and uniform c-convexity of complex normed spaces are introduced as a natural generalization of strict and uniform convexity. It is proved that the complex space L .(S, a, p) is uniformly c-convex. An application to analytic functions is given. Throughout, the open unit disc in the complex plane is denoted by A. Definition 1. A complex normed space X is called strictly c-convex if x, y £ X, \\x\\ = 1 and ||x + Cy\\ 0 there exists 8 > 0 such that x, y £ X, \\x + (Cy\\ e implies ||x|| 0, let (0) = 0 (resp. limgXo&Jc(r5) = 0). It is obvious that strict (resp. uniform) convexity of a complex normed space implies its strict (resp. uniform) c-convexity. On the other hand we show that there exist uniformly c-convex spaces with no extreme points on the unit sphere. Thorp and Whitley (see 16]) have proved that the complex space LxiS, o, p) is strictly c-convex. Here we generalize this by proving that Lj(c, o, p) is uniformly c-convex. Namely, we prove Theorem 1. Let X be the complex space LAS, a, p). Let 8 > 0 and let x, y £ X satisfy \\x\\ = 1, ||x ±y|| < 1 + 8, \\x ±iy\\ < 1 + 8. Then (1) [|y|| < ^(4 + 2(1 + 25)*). Received by the editors April 26, 1973 and, in revised form, December 3, 1973. AMS (MOS) subject classifications (1970). Primary 30A96.

85 citations


"Level sets of the condition spectru..." refers background or methods in this paper

  • ...([9], Remark) Let X be a complex Banach space....

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  • ...Step 2 : In this step we apply Theorem 2 in [9] to the functions ψn and we see the consequence....

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  • ...Applying, Theorem 2 in [9] to the function ψn, we get ‖ψn(λ)− ψn(0)‖≤ ( 2|λ| 1− |λ| ) wc (1− ‖ψn(0)‖) for all λ ∈ B(0, 1)....

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  • ...In [9], Theorem 1, Globevnik showed L1 space is complex uniformly convex....

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