Spectral Radius of a Normal Operator

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.

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 .

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 \(1\le q<2\).

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).

Recall that the spectrum, σ(T), of a linear operator T in a complex Banach space is the set of complex numbers λ such that T—λI does not have a densely defined bounded inverse. It is known [7, § 5.1] that σ(T) is a closed subset of the complex plane C. If T is not bounded, σ(T) may be empty or the whole of C. If σ(T) ≠ C and T is closed the spectral mapping theorem, is valid for complex polynomials p(z) [7, §5.7]. Also, if T is closed and λ ∉ σ(T), (T–λI)−1 is everywhere defined.

Given the abstract evolution equation y ′ ( t ) = A y ( t ) , t ∈ ℝ , y^{\prime} (t)=Ay(t),t\in {\mathbb{R}}, with a scalar type spectral operator A in a complex Banach space, we find conditions on A, formulated exclusively in terms of the location of its spectrum in the complex plane, necessary and sufficient for all weak solutions of the equation, which a priori need not be strongly differentiable, to be strongly Gevrey ultradifferentiable of order β ≥ 1 \beta \ge 1 , in particular analytic or entire, on ℝ {\mathbb{R}} . We also reveal certain inherent smoothness improvement effects and show that if all weak solutions of the equation are Gevrey ultradifferentiable of orders less than one, then the operator A is necessarily bounded. The important particular case of the equation with a normal operator A in a complex Hilbert space follows immediately.

Each bounded linear operator a on a Hilbert space K has a hermitian left-support projection p such that and (1 – p)K = ker α* = ker αα*. I demonstrate here that certain operators on Banach spaces also have left supports.Throughout this paper X will be a complex Banach space with norm-dual X', and L(X) will be the Banach algebra of bounded linear operators on X. Two linear subspaces Y and Z of X are orthogonal (in the sense of G. Birkhoff) if ∥ y ∥ ≦ ∥ y + z ∥ (y ∈Y, z ∈ Z); this orthogonality relation is not, in general, symmetric. It is easy to see that pX is orthogonal to (1 – p)X if and only if the norm of p is 0 or 1, when p is a projection on X.

This note presents an asymptotic formula for the minimum of the moduli of the elements in the spectrum of a bounded linear operator acting on Banach space X. This minimum moduli is called the inner spectral radius, and the formula established herein is an analogue of Gelfand′s spectral radius formula.

Generalizations of Self-Adjointness to Banach Spaces

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 G be a semi-direct product with A Abelian and K compact. We characterize spread-out probability measures on G that are mixing by convolutions by means of their Fourier transforms. A key tool is a spectral radius formula for the Fourier transform of a regular Borel measure on G that we develop, and which is analogous to the well-known Beurling-Gelfand spectral radius formula. For spread-out probability measures on G, we also characterize ergodicity by convolutions by means of the Fourier transform of the measure. Finally, we show that spread-out probability measures on such groups are mixing by convolutions if and only if they are weakly mixing by convolutions.

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

Let be the complex Banach algebra of all bounded linear operators on a complex Banach space E For n-tuples and of operators on E, let R A,B denote the operator on defined by For we put In this note, we prove that where V(·) is the joint spatial numerical range, W 0(·) is the algebraic numerical range and J is a norm ideal of We shall show that this inclusion becomes an equality when R A,B is taken to be a derivation. Also, we deduce that for and J is a norm ideal of , where w(·) is the numerical radius. On the other hand, in the particular case when E is a Hilbert space, we shall prove that the lower estimate bound holds, if one of the following two conditions is satisfied: J is a standard operator algebra of and A,B∈ J. J is a norm ideal of and

The classical Friedrichs extension was generalized by Sauer [Quaestiones Mathematicae, Vol. 16(1989), 239-249] to obtain the simultaneous extension of the operator pair with X a complex Banach space and y a complex Hilbert space. The extension procedure includes the introduction of a third operator C0 for which the form yields a norm in the subspace D0Certain symmetry conditions are also assumed. It was shown that for C0 = B0, the extension could be used to obtain existence and uniqueness results for initial value problems for second order parabolic equations with dynamic boundary values. In the paper it is shown that the restriction of symmetry may be removed and that wider choices of C0 lead to existence and uniqueness results for a larger class of dynamic boundary value problems.

B will always denote a commutative semi-simple Banach algebra with a unit element. If fe B then r(/) denotes its spectral radius. A sequence F=(/, ) is called a spectral null sequence if \\fj\\ ^ 1 for each j, while limy-*,r(fj) = 0. If F=(fj) is a spectral null sequence we put TN(F)limsupj-c \\ff \\ lN for each N S 1. Finally we define the complex number TN(B) -sup {r^CF): F is a spectral null sequence in B). In general r(B) = 1 for all N ^ 1 and the aim of this paper is to study the case when TN(B) < 1 for some N.