Banach Algebras and Spectral Theory for Operators on a Banach Space

2007 ◽  
pp. 187-231
Author(s):  
John B. Conway
Keyword(s):  
Banach Space ◽  
Spectral Theory ◽  
Banach Algebras

scholarly journals Some inequalities for norm unitaries in Banach algebras

1978 ◽  
Vol 21 (1) ◽  
pp. 17-23 ◽  
Author(s):  
M. J. Crabb ◽  
J. Duncan
Keyword(s):  
Banach Space ◽  
Banach Algebra ◽  
Unit Circle ◽  
Banach Algebras ◽  
Bounded Operators ◽  
Image Position ◽  
Unital Banach Algebra

Let A be a complex unital Banach algebra. An element u∈A is a norm unitary if(For the algebra of all bounded operators on a Banach space, the norm unitaries arethe invertible isometries.) Given a norm unitary u∈A, we have Sp(u)⊃Γ, where Sp(u) denotes the spectrum of u and Γ denotes the unit circle in C. If Sp(u)≠Γ we may suppose, by replacing eiθu, that . Then there exists h ∈ A such that

The spectrum and eigenspaces of a meromorphic operator-valued function

1997 ◽  
Vol 127 (5) ◽  
pp. 1027-1051 ◽  
Author(s):  
Robert Magnus
Keyword(s):  
Banach Space ◽  
Riemann Surface ◽  
Spectral Theory ◽  
Approximation Theorem ◽  
Meromorphic Mapping ◽  
Bounded Operators ◽  
Single Operator ◽  
Definition Of ◽  
Vector Valued Functions ◽  
Vector Valued

SynopsisIt is shown how to associate eigenvectors with a meromorphic mapping defined on a Riemann surface with values in the algebra of bounded operators on a Banach space. This generalises the case of classical spectral theory of a single operator. The consequences of the definition of the eigenvectors are examined in detail. A theorem is obtained which asserts the completeness of the eigenvectors whenever the Riemann surface is compact. Two technical tools are discussed in detail: Cauchy-kernels and Runge's Approximation Theorem for vector-valued functions.

scholarly journals Spectral integration and spectral theory for non-Archimedean Banach spaces

2002 ◽  
Vol 31 (7) ◽  
pp. 421-442 ◽  
Author(s):  
S. Ludkovsky ◽  
B. Diarra
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Banach Algebras ◽  
Linear Operators ◽  
Spectral Integration ◽  
Different Types ◽  
Commutative Subalgebras ◽  
Continuous Operators ◽  
Stone Theorem ◽  
Continuous Linear Operators

Banach algebras over arbitrary complete non-Archimedean fields are considered such that operators may be nonanalytic. There are different types of Banach spaces over non-Archimedean fields. We have determined the spectrum of some closed commutative subalgebras of the Banach algebraℒ(E)of the continuous linear operators on a free Banach spaceEgenerated by projectors. We investigate the spectral integration of non-Archimedean Banach algebras. We define a spectral measure and prove several properties. We prove the non-Archimedean analog of Stone theorem. It also contains the case ofC-algebrasC∞(X,𝕂). We prove a particular case of a representation of aC-algebra with the help of aL(Aˆ,μ,𝕂)-projection-valued measure. We consider spectral theorems for operators and families of commuting linear continuous operators on the non-Archimedean Banach space.

scholarly journals Amenability of ultrapowers of Banach algebras

2009 ◽  
Vol 52 (2) ◽  
pp. 307-338 ◽  
Author(s):  
Matthew Daws
Keyword(s):  
Banach Space ◽  
Reflexive Banach Space ◽  
Banach Algebras ◽  
Tensor Products ◽  
Convolution Algebra ◽  
Abstract Characterization ◽  
Closed Subalgebra

AbstractWe study when certain properties of Banach algebras are stable under ultrapower constructions. In particular, we consider when every ultrapower of $\mathcal{A}$ is Arens regular, and give some evidence that this is so if and only if $\mathcal{A}$ is isomorphic to a closed subalgebra of operators on a super-reflexive Banach space. We show that such ideas are closely related to whether one can sensibly define an ultrapower of a dual Banach algebraffi We study how tensor products of ultrapowers behave, and apply this to study the question of when every ultrapower of $\mathcal{A}$ is amenable. We provide an abstract characterization in terms of something like an approximate diagonal, and consider when every ultrapower of a C*-algebra, or a group L1-convolution algebra, is amenable.

scholarly journals Strictly Convex Banach Algebras

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 221
Author(s):  
David Yost
Keyword(s):  
Banach Space ◽  
Banach Algebras ◽  
Infinite Dimensions ◽  
Strictly Convex ◽  
C Algebras ◽  
Finite Dimensional ◽  
And Algebra ◽  
Open Question

We discuss two facets of the interaction between geometry and algebra in Banach algebras. In the class of unital Banach algebras, there is essentially one known example which is also strictly convex as a Banach space. We recall this example, which is finite-dimensional, and consider the open question of generalising it to infinite dimensions. In C*-algebras, we exhibit one striking example of the tighter relationship that exists between algebra and geometry there.

scholarly journals A characterisation and two examples of Riesz operators

1968 ◽  
Vol 9 (2) ◽  
pp. 106-110 ◽  
Author(s):  
T. A. Gillespie ◽  
T. T. West
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Linear Operator ◽  
Spectral Theory ◽  
Bounded Linear Operator ◽  
Spectral Projection ◽  
Resolvent Operators ◽  
Riesz Operator ◽  
Bounded Linear ◽  
Riesz Operators

A Riesz operator is a bounded linear operator on a Banach space which possesses a Riesz spectral theory. These operators have been studied in [5] and [6]. In §2 of this paper we characterise Riesz operators in terms of their resolvent operators. In [6] it was shown that every Riesz operator on a Hilbert space can be decomposed into the sum of compact and quasi-nilpotent parts. §3 contains an example to show that these parts cannot, in general, be chosen to commute. In §4 the eigenset of a Riesz operator is defined. It is a sequence of quadruples each of which consists of an eigenvalue, the corresponding spectral projection, index and nilpotent part. This sequence satisfies certain obvious conditions, and the question arises of the existence of a Riesz operator which has such a sequence as its eigenset. We give an example of an eigenset which has no corresponding Riesz operator.

scholarly journals Approximate complementation and its applications in studying ideals of Banach algebras

2003 ◽  
Vol 92 (2) ◽  
pp. 301 ◽  
Author(s):  
Yong Zhang
Keyword(s):  
Banach Space ◽  
Banach Algebra ◽  
Compact Subset ◽  
Approximation Property ◽  
Banach Algebras ◽  
Approximate Identity ◽  
Amenable Banach Algebra

We show that a subspace of a Banach space having the approximation property inherits this property if and only if it is approximately complemented in the space. For an amenable Banach algebra a closed left, right or two-sided ideal admits a bounded right, left or two-sided approximate identity if and only if it is bounded approximately complemented in the algebra. If an amenable Banach algebra has a symmetric diagonal, then a closed left (right) ideal $J$ has a right (resp. left) approximate identity $(p_{\alpha})$ such that, for every compact subset $K$ of $J$, the net $(a\cdot p_{\alpha})$ (resp. $(p_{\alpha}\cdot a)$) converges to $a$ uniformly for $a \in K$ if and only if $J$ is approximately complemented in the algebra.

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