Semi-groups and collectively compact sets of linear operators

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).

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 .

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 { T t } \{ {T_t}\} be a strongly continuous semigroup of bounded linear operators on a Banach space X X , satisfying lim t → ∞ | | T t | | / t = 0 {\lim _{t \to \infty }}||{T_t}||/t = 0 . We prove the equivalence of the following conditions: (1) t − 1 ∫ 0 t T r d r {t^{ - 1}}\int _0^t {{T_r}dr} converges uniformly as t → ∞ t \to \infty . (2) The infinitesimal generator A A has closed range. (3) lim λ → 0 + λ R λ {\lim _{\lambda \to {0^ + }}}\lambda {R_\lambda } exists in the uniform operator topology.

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.

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.

We study the well-posedness of the fractional differential equations with finite delay: Dαu(t) + BDβu(t) = Au(t) + Fut + f(t), (0 ≤ t ≤ 2π) on Lebesgue-Bochner spaces and periodic Besov spaces , where A and B are closed linear operators in a complex Banach space X satisfying D(A) ⊂ D(B), 0 ≤ β < α, α ≥ 1 and F is a bounded linear operator from Lp ([−2π, 0]; X) (resp. into X. We give necessary and sufficient conditions for the Lp -well-posedness and the -well-posedness of above equations by using known operator-valued Fourier multiplier theorems.

In this survey we report on very recent results about some non-linear geometrical properties of many classes of real and complex Banach spaces and uniform algebras, including the ball algebra \(\fancyscript{A}_u(B_X)\) of all uniformly continuous functions on the closed unit ball and holomorphic on the open unit ball of a complex Banach space \(X\). These geometrical properties are: Polynomial numerical index, Polynomial Daugavet property and Bishop-Phelp-Bollobas property for multilinear mappings.

Let E be a complex Banach space, with the unit ball B E . We study the spectrum of a bounded weighted composition operator u C φ on H ∞(B E ) determined by an analytic symbol φ with a fixed point in B E such that φ(B E ) is a relatively compact subset of E, where u is an analytic function on B E .

In this paper, we establish the sharp Fekete–Szegö inequalities of functional type over a new class of holomorphic mappings on the unit ball in a complex Banach space. This class includes some important widely studied classes. We offer simpler proofs and require less restrictive assumptions than those found in recent literature. Finally we show that some important previous results follows from ours by choosing suitable parameters.

Numerical peak holomorphic functions on Banach spaces

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 \(1\le q<2\).

Let be a linear operator with domain in a complex Banach space . An element is called a vector of degree at most relative to if , . The set of vectors of degree at most is denoted by and the least deviation of an element of from the set is denoted by . For a fixed sequence of positive numbers consider a function . Conditions for the sequence and the operator are found that ensure the equality for . If the quantity on the left-hand side of this formula is finite, then belongs to the Hadamard class determined by the operator and the sequence . One consequence of the above formula is an expression in terms of for the radius of holomorphy of the vector-valued function , where , and is an entire function.

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 starting point for this investigation was an attempt to generalize the well known theorem of Silverman and Toeplitz [25](1) on regularity of a sequence-to-sequence transformation. This theorem may be stated as follows: If a transformation of sequences { t, } of real numbers to sequences { Sm } is defined from a matrix { amn }, m, n = 1, 2, * , by the equations Sm =ZEn amntn, the transformation is regular-that is, is defined everywhere and takes every convergent sequence { t, } into another convergent sequence with the same limit-if and only if the matrix { amn } satisfies the conditions (a) limm dn 'tamn = 1, (b) limm amn = 0 for each n, and (c) there is a K such that En I amn, < K for every m. In the special case under consideration, the fact that regularity implies condition (c) (the non-trivial part of the proof) can be derived from a theorem of Banach [3, p. 80, Theorem 5 ]: If A and B are Banach spaces, and if U, n = 1, 2, * * , are. linear operators on A to B, such that lim supn || Un(a)ll < o? for each a in A, then lim supn 11 U711 < oo If the sequence of integers is replaced by a directed set X, it is known that A, B, X, and U. can be chosen for which the similar statement relating lim supx || Ux(a)ll and lim supx || Ux,l is false; sections 1-3 of this paper consider these cases in an attempt to solve the problem of boundedness: Characterize those Banach spaces A and B, and directed sets X such that choosing the linear operators U, on A to B so that lim supx || U,(a)ll < oo for each a in A implies that lim supx || Ux|| < oo. Section 1 is a review of pertinent facts about directed sets and convergence (mostly due to Moore and Smith [19], G. Birkhoff [5], and Tukey [27]). Section 2 studies the relations among three topologies in the space of operators on A to B. In ?3 the problem of boundedness is studied but not completely solved. The second part of the paper is concerned with certain special operators on some function spaces. In ?4 the space is that of the totally measurable functions on a set Y to a Banach space B; a class of operators on this space is defined in terms of additive, real-valued set-functions and the relations among various topologies in this set of operators is given; this is used in ?6 to give a general form to a theorem of Vulich [28]. In ?5 the functions studied are the measurable functions on Y to B; the operations on this space are defined in terms of completely additive, limited, set-functions whose values are transfor-