An unbounded spectral mapping theorem

Let Σ denote the set of all seriesof complex numbers. By a ‘summability method’, say A, we mean a function from some subset (the set of ‘A -summable series’) of Σ into the set of complex numbers. We will use the language generally associated with this definition, and will take for granted the case in which A is (C, 1). A summability method A will be called linear if, whenever a, b are A -summable, then so is λa + μb (where λ, μ are any complex constants) and if the. A -sums of a, b, λa + μb are then related in the natural way. We call A regular if, whenever a converges to σ, it is A -summable to σ. If A is a regular summability method, then any condition P on the series (1) will be called a Tauberian condition for A if any A -summable series which satisfies P is convergent.

K-POTENT PRESERVING LINEAR MAPS

Let E be a complex Banach space and F be a complex Banach algebra. We will be interested in the subspace ℙ_f({}^n E,F) of P({}^n E,F) generated by the collection of functions \varphi^n(n\in ℕ, \varphi \in L(E,F)) where \varphi^n(x) = (\varphi (x))^n for each x\in E .

It is shown that in every complex Banach space E with open unit ball D C E there is a closed C-linear subspace Cc such that V C\ D is the orbit of the origin 0 E E under the group Aut(D) of all biholomorphic automorphisms of D. In particular two complex Banach spaces are isometrically equivalent if and only if their open unit balls are biholomorphically equivalent.Suppose E, F are complex Banach spaces and U E E is an open subset.Then a mapping/: U -* Fis called holomorphic if the Frchet derivative of/ at x (denoted by f'(x)) exists for every x G U. (For details compare [6].)In this paper we show that the Banach space E is completely determined by the holomorphic structure of its (open) unit ball; more precisely, Theorem.The complex Banach spaces E and are isometrically isomorphic if and only if there is a biholomorphic mapping from the unit ball of E onto the unit ball of .This answers a question L. A. Harris raised in [3, p. 15].For the proof let be a fixed complex Banach space with unit ball DEE.Let 0 denote the set of all holomorphic vector fields on D i.e. of all expressions X = f(z)d/dz where/: D -* E is a holomorphic mapping and z denotes the coordinate of D. Then D is a complex Lie algebra in the product [/W,^)]=(.'W/(z)-/'(z)g(z)).For every integer v the Lie algebra D contains the linear subspace ^v = l h(z)~~ h: E ^> E is a. homogeneous polynomial of degree v + 1 >.Obviously [$r,$J C SJL^ for every v and \i.Since D is a bounded domain in E, the set g of all complete holomorphic vector fields on D has the following properties (compare [7]):(0) Every h(z)d/dz G g is uniquely determined by h(0) and h'(0) (Cartan's uniqueness theorem).

It has long been customary in the theory of linear transformations-integral equations, linear equations in an infinite number of unknowns, etc.-to make use of certain notions from the theory of functions of a complex variable. In particular, the well known Neumann expansion of the resolvent is a formal power series whose coefficients, instead of being numbers, are iterates of a certain linear transformation.' These resemblances to the classical CauchyWeierstrass theory can be described accurately in terms of a theory of analytic functions whose values lie in a complex Banach space. The possibility of such a theory, derived in the usual manner from Cauchy's theorem and integral formula, was recognized by Wiener.2 Still further generalizations, in which the independent variable may also range over a complex Banach space, have been considered by a number of people.3 It therefore seems desirable to outline a general abstract theory of linear operations depending analytically on a parameter, which we shall here suppose to be a complex variable. DEFINITION 1.1 A function f(X) defined in an open set A of the complex plane, with values in a complex Banach space E, is called analytic in A if the limit

Let X be a Complex Banach space and T be a bounded operator in X. The number sup {|λ| : λ ∈ σ(T)} (where σ(T) is the spectrum of T and σ(T) ̸= ϕ) is called the spectral radius of T and denoted by r(T). Since λ ≤ ∥T∥for all λ ∈ σ(T), it follows that r(T) ≤ ∥T∥. The spectral mapping theorem implies that r(Tn) = (r(T))n for every positive integer n. It frequently turns out that it is easy to compute the spectral radius of an operator even if it is hard to _nd the spectrum. This is often made easy by the spectral radius formula. Let H be a Hilbert space and T be a bounded linear operator in H. In this paper we show that if T is normal, then Tn is normal for each n ∈ N and ∥Tn∥ = ∥T∥n. Consequently, we use the spectral radius formula to show that r(T) = ∥T∥. Moreover, we show that if X is a Complex Banach space and T is bounded in X then there is a λ belonging to the spectrum of T such that |λ| = r(T). Let H be a Complex Hilbert space and T be a bounded operator in H which is normal; we show that ∥T∥ = sup {|Tx, x| : x ∈ H and ∥x∥ = 1} and the residual spectrum of T is void.

This note is a review of the paper of the authors [8]. In this paper, nondegenerately ultraprime normed Jordan algebras are introduced and studied under the light of Zel’manov’s prime theorem. With this aim, ultra-*-prime normed associative algebras are introduced and, for such an algebra A, a normed version of the symmetric Martindale ring of quotients of A (denoted by Q b (A))is provided. The main result asserts that, up to bicontinuous isomorphisms, the non-degenerately ultraprime Jordan-Banach algebras are the following: the only simple exceptional 27-dimensional Jordan complex algebra, the Jordan-Banach algebras of a regular continuous non-degenerate symmetric bilinear form on a complex Banach space, and the closed Jordan subalgebras of Q b (A) contained in H(Q b (A), *) and containing H(A, *) as an ideal, where A is an ultra-*-prime associative complex Banach algebra such that H(A, *) generates A as a Banach algebra.

A set of linear operators from one Banach space to another is collectively compact if and only if the union of the images of the unit ball has compact closure. Semi-groups S = {T(t): t ^ 0} of bounded linear operators on a complex Banach space into itself and in which every operator T(t), t > 0 is compact are considered. Since Tit, + t 2) = T(tJT(tJ for each operator in the semi-group, it would be expected that the theory of collectively compact sets of linear operators could be profitably applied to semi-groups. 1* Introduction* Let X be a complex Banach space with unit ball Xλ and let [X, X] denote the space of all bounded linear operators on X equipped with the uniform operator topology. The semi-group definitions and terminology used are those of Hille and Phillips [6]. Let S be a semi-group of vector-valued functions T: [0, oo)—>[X, X]. It is assumed that T(t) is strongly continuous for t ^ 0. If lim^ || T(t)x - T(to)x || = 0 for each t0 ^ 0, xeX and if there is a constant M such that the || T(t) || ^ M for each t ^ 0, then S — {T(t): t ^> 0} is called an equicontinuou s semi-group of class Co. The infinitesimal generator A of the semi-group S is defined by Ax = lim— [T(s)x - x] s->o S whenever the limit exists. The domain D(A) of A is a dense subset of X consisting of just those elements x for which this limit exists. A is a closed linear operator having resolvents R(X) which, for each complex number λ with the real part of λ greater than zero, are given by the absolutely summable Riemann-Stieltjes integral

The multiplicities of the powers of a bounded linear operator T T , acting on a complex separable infinite-dimensional Banach space X \mathfrak {X} , satisfy the inequalities \[ ( ∗ ∗ ) μ ( T n ) ⩽ μ ( T h n ) ⩽ h μ ( T n ) for all h , n ⩾ 1. ( * * )\qquad \mu ({T^n}) \leqslant \mu ({T^{hn}}) \leqslant h\mu ({T^n})\quad {\text {for}}\;{\text {all}}\;h,n \geqslant 1. \] Nothing else can be said, in general, because simple examples show that for each sequence { μ n } n = 1 ∞ \{ {\mu _n}\} _{n = 1}^\infty , satisfying the inequalities ( ∗ ∗ ) ( * * ) , there exists T T acting on X \mathfrak {X} such that μ ( T n ) = μ n \mu ({T^n}) = {\mu _n} for all n ⩾ 1 n \geqslant 1 .

Centralizers in semisimple algebras, and descent spectrum in Banach algebras

Much has been learned in recent years about the existence, determination, and properties of power-series-like expansions for expressing a nonlinear system's outputs in terms of its inputs. In particular, the existence and local convergence of expansions, and of certain “associated expansions,” for important large classes of systems are now well established. While the focus of attention has been on questions such that the size of the inputs for which convergence is guaranteed is not the main issue, some related material has appeared that bears on the problem of determining the extent of the region of convergence. The result most closely related to this paper is a recent theorem that gives necessary and sufficient conditions under which f <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−1</sup> has a generalized power-series expansion when f is an invertible locally-Lipshitz map between certain general subsets of two complex Banach spaces. In applications involving nonlinear models, ordinarily only real spaces of inputs and outputs are of direct interest. A “complexification” involving a certain solvability condition in complex spaces has to be able to be carried out to use the theorem referred to above. This paper reports on pertinent general results concerning invertible maps between subsets of real Banach spaces, with their complex extensions, and with generalized power-series expansions in both real and complex spaces. It focuses on questions concerning expansions for inverses of maps defined in real spaces. The results show that for a very large class of systems that have input/output maps, the ability to complexify is not just a useful sufficient condition for expandability, but is in fact the key condition for an input/output map to be representable by a generalized power-series expansion.

Cartan–Slodkowski spectra, splitting elements and noncommutative spectral mapping theorems

We prove that the uniform growth bound of a discrete evolution family of bounded linear operators acting on a complex Banach space X satisfies the inequalityhere is the operator norm of a convolution operator which acts on a certain Banach space of X-valued sequences.

A bounded linear operator $T$ on a complex Banach space $\mathcal{X}$ is said to be full if $\overline{T\mathcal{M}}=\mathcal{M}$ for every invariant subspace $\mathcal{M}$ of $\mathcal{X}$. It is nearly full if $\overline{T\mathcal{M}}$ has finite codimension in $\mathcal{M}$. In this paper, we focus our attention to characterize full and nearly full operators in complex Banach spaces, showing that some valid results in complex Hilbert spaces can be generalized to this context.

Following the scalar-valued case considered by Djakow and Ramanujan (A remark on Bohr’s theorem and its generalizations 14:175–178, 2000) we introduce, for each complex Banach space X and each \(1\le p<\infty \), the p-Bohr radius of X as the value $$\begin{aligned} r_p(X)= \sup \bigg \{r\ge 0: \sum _{n=0}^\infty \Vert x_n\Vert ^pr^{np}\le \sup _{|z|<1} \Vert f(z)\Vert ^p\bigg \} \end{aligned}$$ where \(x_n\in X\) for each \(n\in \mathbb {N}\cup \{0\}\) and \(f(z)=\sum _{n=0}^\infty x_nz^n\in H^\infty (\mathbb {D},X)\). We show that a complex (possibly infinite dimensional) Banach space X is p-uniformly \(\mathbb {C}\)-convex for \(p\ge 2\) if and only if \(r_{p}(X)>0\). We study the p-Bohr radius of the Lebesgue spaces \(L^q(\mu )\) for different values of p and q. In particular we show that \(r_p(L^q(\mu ))=0\) whenever \(p<2\) and \(dim(L^q(\mu ))\ge 2\) and \(r_p(L^q(\mu ))=1\) whenever \(p\ge 2\) and \(p'\le q\le p\). We also provide some lower estimates for \(r_2(L^q(\mu ))\) for the values \(1\le q<2\).