Generalized inverses over Banach algebras

As a continuation of the paper J. Math. Anal. Appl. 464(2018), 939-954. We will define and examine the $(n,\varepsilon)$-pseudospectra and condition $(n,\varepsilon)$-pseudospectra of an element pencil in a unital complex Banach algebra. We characterize the $(n,\varepsilon)$-pseudospectra of an element pencil in a unital complex Banach algebra. In particular, we demonstrate some properties of $(n,\varepsilon)$-pseudospectra and condition $(n,\varepsilon)$-pseudospectra of an element pencil in unital complex Banach algebras.

We show that a real Banach algebra $A$ such that $\Vert a^{2}\Vert =\Vert a\Vert^{2}$ for $a\in A$ is a subalgebra of the algebra $C_{\mathbb{H}}( X) $ of continuous quaternion-valued functions on a compact set $X$.

For a matrix over a complex commutative unital Banach algebra, necessary and sufficient conditions are given for the existence of its group inverse, and more generally, its Drazin inverses. The conditions are easy to check and explicit formulas for the inverses are provided. Some properties of the inverses and an application to operator theory are discussed. This note is a continuation of an earlier work of the author.

Let A be a complex unital Banach algebra. Since the set of invertible elements is open, there is an open ball around every invertible element. In this article, we investigate the Banach algebras for which the radius given by the Neumann series is optimal.

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

We show that for any semisimple, commutative, complex Banach algebra $A$ with unit there are norms on $A$, which we call natural norms, equivalent to the original norm on $A$ with the following property: Let $(A,|| \cdot |{|_A},{e_A})$ and $(B,|| \cdot |{|_B},{e_B})$ are commutative, semisimple Banach algebras with units and natural norms. Assume $T$ is a linear isometry from $(A,|| \cdot |{|_A})$ onto $(B,|| \cdot |{|_B})$ with $T{e_A} = {e_B}$. Then $T$ is an isomorphism in the category of Banach algebras. For a fairly large class of algebras, for example, for uniform algebras, for algebras of the form ${C^k}(X),{\text { Lip}}(X),{\text { AC}}(X)$, the natural norm we have defined coincides with a usual norm.

In this paper we derive the explicit, closed-form, recursion-free formulae for the arbitrary-order Fr?chet derivatives of the exponential and logarithmic functions in unital Banach algebras (complex or real). These computations are obtained via the Bochner integrals for the Banach algebra valued functions, with respect to the standard Lebesgue measure. As an application, we utilize our results in the approximation schemes of the solutions to stochastic functional differential equations.

Let A A , B B be bounded operators on a Banach space with 2 π i 2\pi i -congruence-free spectra such that e A e B = e B e A e^Ae^B=e^Be^A . E. M. E. Wermuth has shown that A B = B A AB=BA . Ch. Schmoeger later established this result, using inner derivations and, in a second paper, has shown that: for a , b a,b in a complex unital Banach algebra, if the spectrum of a + b a+b is 2 π i 2\pi i -congruence-free and e a e b = e a + b = e b e a e^ae^b=e^{a+b}=e^be^a , then a b = b a ab=ba (and thus, answering an open problem raised by E. M. E. Wermuth). In this paper we use the holomorphic functional calculus to give alternative simple proofs of both of these results. Moreover, we use the Borel functional calculus to give new proofs of recent results of Ch. Schmoeger concerning normal operator exponentials on a complex Hilbert space, under a weaker hypothesis on the spectra.

According to Kordula and Muller [5] and Muller [6, p. 50], a regularity in a complex unital Banach algebra A is defined as a set 1Z C ^4, such that (i) if a G A and n G N, then a G K ^=> an G Tl' (ii) if a, b are relatively prime elements of A, then ab G 1Z « * a G 1Z and b G 1Z (a, e are relatively prime if there exist x,y G A such that {a, 6, x, y} is a commuting set and ax + by = 1). The concept of regularity is at the heart of the Kordula-Muller axiomatic spectral theory, where the spectrum of a G A relative to the regularity 7Z is defined by:

Let \(\mathcal{A}\) be a complex unital Banach algebra. An element a ∈ \(\mathcal{A}\) is said to be Hermitian if ‖exp(ita)‖ = 1 for all t ∈ ℝ. In the case of the algebra of bounded linear operators in a Hilbert space, this Hermitian property agrees with the ordinary self-adjointness. If a ∈ \(\mathcal{A}\) is Hermitian, then ‖a‖ = |a|, where |a| denotes the spectral radius of a. A function F: ℝ → ℂ is called a universal symbol if ‖F(a)‖ = | F(a)| for every \(\mathcal{A}\) and all Hermitian a ∈ \(\mathcal{A}\). We characterize universal symbols in terms of positive-definite functions.

We present here a multiplicative version of the classical Kowalski–Słodkowski theorem, which identifies the characters among the collection of all functionals on a complex and unital Banach algebra A. In particular, we show that, if A is a $C^\star $ -algebra, and if $\phi :A\to \mathbb C $ is a continuous function satisfying $ \phi (x)\phi (y) \in \sigma (xy) $ for all $x,y\in A$ (where $\sigma $ denotes the spectrum), then either $\phi $ is a character of A or $-\phi $ is a character of A.

Invariants of algebraic derivations and automorphisms in Banach algebras

In the eighties of the last century Rennison studied and characterized the set of quasi-central elements of some unital Banach algebras. In this paper, we improve one of Rennison's result assert that for given unital Banach algebra \(A\) the set of quasi-central elements \(Q(A)\), need not be equal to centre of \(A\) and need not be closed under addition or multiplication. Further, we prove that if \(A\) is unital ultraprime Banach algebra, then \(Q(A)\) need not be equal to centre of \(\mathrm{A}\) and need not be closed under addition or multiplication.

In this paper, we present some results concerning spectral properties of multipliers on commutative semi-simple Banach algebras. The study concentrates on multipliers whose (local) spectra are in a sense natural. Particular emphasis is given to the determination of the (local) spectra of multipliers in connection with Beurling algebras, Dunford property (C) and Hille-Yosida space.

Uniqueness under spectral variation in the socle of a Banach algebra