Complex Banach Space Research Articles

On the hereditary character of certain spectral properties and some applications

In this paper we study the behavior of certain spectral properties of an operator T on a proper closed and T-invariant subspace W ⊆ X such that Tn (X) ⊆ W, for some n ≥ 1, where T ∈ L(X) and X is an infinite-dimensional complex Banach space. We prove that for these subspaces a large number of spectral properties are transmitted from T to its restriction on W and vice-versa. As consequence of our results, we give conditions for which semiFredholm spectral properties, as well as Weyl type theorems, are equivalent for two given operators. Additionally, we give conditions under which an operator acting on a subspace can be extended on the entire space preserving the Weyl type theorems. In particular, we give some applications of these results for integral operators acting on certain functions spaces.

Superconvex-cyclicity of operators

A bounded linear operator T on a complex Banach space is superconvex-cyclic, if for some vector x in the space, is dense. Here, means the convex hull of . In this paper, we give necessary conditions for the superconvex-cyclicity of an operator in terms of the point spectrum of its adjoint. Also, we prove that positive multiples of superconvex-cyclic operators are superconvex-cyclic. However, this concept is not necessarily transferred to positive integer powers of an operator. Moreover, we completely characterize superconvex-cyclic matrices whose spectra contain no real number or contain a positive real number.

Multiplicative perturbations of local C-cosine functions

"We establish some left and right multiplicative perturbations of a local C-cosine function C(.) on a complex Banach space X with non-densely defined generator, which can be applied to obtain some new additive perturbation results concerning C(.)."

Local commutants and ultrainvariant subspaces

For an operator A on a complex Banach space X and a closed subspace M⊆X, the local commutant of A at M is the set C(A;M) of all operators T on X such that TAx=ATx for every x∈M. It is clear that C(A;M) is a closed linear space of operators, however it is not an algebra, in general. For a given A, we show that C(A;M) is an algebra if and only if the largest subspace MA of X such that C(A;M)=C(A;MA) is invariant for every operator in C(A;M). We say that a closed subspace M⊆X is ultrainvariant for A∈B(X) if it is invariant for every operator in C(A;M). For several types of operators we prove that they have non-trivial ultrainvariant subspaces. For a normal operator on a Hilbert space, every hyperinvariant subspace is ultrainvariant. On the other hand, the family of all ultrainvariant subspaces of a nilpotent operator can be a proper subset of the lattice of all hyperinvariant subspaces.

(ω, c)-periodic and asymptotically (ω, c)-periodic mild solutions to fractional Cauchy problems

In this paper, we establish some new properties of -periodic and asymptotically -periodic functions, then we apply them to study the existence and uniqueness of mild solutions of these types to the following semilinear fractional differential equations: (1) and (2) where stands for the Caputo derivative and A is a linear densely defined operator of sectorial type on a complex Banach space and the function is -periodic or asymptotically -periodic with respect to the first variable. Our results are obtained using the Leray–Schauder alternative theorem, the Banach fixed point principle and the Schauder theorem. Then we illustrate our main results with an application to fractional diffusion-wave equations.

Algebras of symmetric analytic functions on Cartesian powers of Lebesgue integrable in a power $p\in [1,+\infty)$ functions

The work is devoted to the study of Fréchet algebras of symmetric (invariant under the composition of every of components of its argument with any measure preserving bijection of the domain of components of the argument) analytic functions on Cartesian powers of complex Banach spaces of Lebesgue integrable in a power $p\in [1,+\infty)$ complex-valued functions on the segment $[0,1]$ and on the semi-axis. We show that the Fréchet algebra of all symmetric analytic entire complex-valued functions of bounded type on the $n$th Cartesian power of the complex Banach space $L_p[0,1]$ of all Lebesgue integrable in a power $p\in [1,+\infty)$ complex-valued functions on the segment $[0,1]$ is isomorphic to the Fréchet algebra of all analytic entire functions on $\mathbb C^m,$ where $m$ is the cardinality of the algebraic basis of the algebra of all symmetric continuous complex-valued polynomials on this Cartesian power. The analogical result for the Fréchet algebra of all symmetric analytic entire complex-valued functions of bounded type on the $n$th Cartesian power of the complex Banach space $L_p[0,+\infty)$ of all Lebesgue integrable in a power $p\in [1,+\infty)$ complex-valued functions on the semi-axis $[0,+\infty)$ is proved.

Wild dynamics and asymptotically separated sets

Let X X be a separable infinite dimensional (real or complex) Banach space. Hájek and Smith, in 2010, constructed a linear bounded operator T T on X X such that A T ≔ { x ∈ X : ‖ T n x ‖ → ∞ } A_T≔\{x\in X:~ \|T^nx\|\to \infty \} is not dense and has nonempty interior, whenever X X admits a symmetric basis. Augé, in 2012, extended the previous construction to each separable Banach space, introducing the notion of wild operators. This work is divided in two parts. In the first part we define and explore the notion of asymptotically separated sets on Banach spaces, giving several examples whenever the ambient space has either finite or infinite dimension. We show how these sets can be used to construct operators with non-intuitive dynamics. Specifically, operators T : X → X T:X\rightarrow X for which the set A T A_T and the set of recurrent points form a partition of the space. Secondly, we investigate geometric and spectral properties of operators with wild dynamic and we provide a wild operator whose spectrum is equal to the closed unit disk. We end this paper giving some comments about the norm closure of the set of wild operators.

Local spectrum, local spectral radius, and growth conditions

Let X be a complex Banach space and \(x\in X.\) Assume that a bounded linear operator T on X satisfies the condition $$\begin{aligned} \left\| e^{tT}x\right\| \le C_{x}\left( 1+\left| t\right| \right) ^{\alpha }\quad \left( \alpha \ge 0\right) , \end{aligned}$$for all \(t\in {\mathbb {R}} \) and for some constant \(C_{x}>0.\) For the function f from the Beurling algebra \(L_{\omega }^{1}\left( {\mathbb {R}} \right) \) with the weight \(\omega \left( t\right) =\left( 1+\left| t\right| \right) ^{\alpha },\) we can define an element in X, denoted by \(x_{f}\), which integrates \(e^{tT}x\) with respect to f. We present a complete description of the elements \(x_{f}\) in the case when the local spectrum of T at x consists of one point. In the case \(0\le \alpha <1,\) some estimates for the norm of Tx via the local spectral radius of T at x are obtained. Some applications of these results are also given.

The Fekete-Szegö problem for starlike mappings and nonlinear resolvents of the Carathéodory family on the unit balls of complex Banach spaces

The Fekete-Szegö problem for starlike mappings and nonlinear resolvents of the Carathéodory family on the unit balls of complex Banach spaces

Analytic bundle structure on the idempotent manifold

Let X be a (real or complex) Banach space (not necessarily a Hilbert space), and $$\mathcal {I}(X)$$ be the set of all non-trivial idempotents; i.e., bounded linear operators on X whose squares equal themselves. We show that, when equipped with the Banach submanifold structure induced from $$\mathcal {L}(X)$$ , the subset $$\mathcal {I}(X)$$ is a locally trivial analytic affine-Banach bundle over the Grassmann manifold $$\mathscr {G}(X)$$ , via the map $$\kappa $$ that sends $$Q\in \mathcal {I}(X)$$ to Q(X), such that the affine-Banach space structure on each fiber is the one induced from $$\mathcal {L}(X)$$ . Using this, we show that if H is a real or complex Hilbert space, then the assignment $$\begin{aligned} (E,T)\mapsto T^*\circ P_{E^\bot } + P_{E}, \quad \text {where}\quad E\in \mathscr {G}(H)\quad \text {and}\quad T\in \mathcal {L}(E,E^\bot ), \end{aligned}$$ induces a real bi-analytic bijection from the total space of the tangent bundle, $$\mathbf {T}(\mathscr {G}(H))$$ , of $$\mathscr {G}(H)$$ onto $$\mathcal {I}(H)$$ (here, $$E^\bot $$ is the orthogonal complement of E, $$P_E\in \mathcal {L}(H)$$ is the orthogonal projection onto E, and $$T^*$$ is the adjoint of T). Notice that this real bi-analytic bijection is an affine map on each tangent plane. Furthermore, if for every $$E\in \mathscr {G}(H)$$ , we identify $$\mathcal {L}(E,E^\bot )$$ with a subspace of $$\mathcal {L}(H)$$ via the embedding $$S\mapsto S\circ P_E$$ , then the inclusion map from $$\mathbf {T}(\mathscr {G}(H))$$ to the trivial Banach bundle $$\mathscr {G}(H)\times \mathcal {L}(H)$$ is a real analytic immersion. Through this, we give a concrete idempotent in $$M_{n^2}\big (C(\mathscr {G}(\mathbb {K}^n))\big )$$ that represents the K-theory class of the tangent bundle $$\mathbf {T}(\mathscr {G}(\mathbb {K}^n))$$ , when $$\mathbb {K}$$ is either the real field or the complex field.

The Birkhoff-James orthogonality and norm attainment for multilinear maps

Very recently, motivated by the result of Bhatia and Šemrl which characterizes the Birkhoff-James orthogonality of operators on a finite dimensional Hilbert space in terms of norm attaining points, the Bhatia-Šemrl property was introduced. The main purpose of this article is to study the denseness of the set of multilinear maps with the Bhatia-Šemrl property which is contained in the set of norm attaining ones. Contrary to the most of previous results which were shown for operators on real Banach spaces, we prove the denseness for multilinear maps on some complex Banach spaces. We also show that the denseness of operators does not hold when the domain space is c0 for arbitrary range. Moreover, we find plenty of Banach spaces Y such that only the zero operator has the Bhatia-Šemrl property in the space of operators from c0 to Y.

On functional equations related to generalized inner derivations on standard operator algebras

Our aim is to prove the following result. Let X be a real or complex Banach space, let ℒ(X) be the algebra of all bounded linear operators on X and let 𝒜(X)⊆ℒ(X) be a standard operator algebra, which possesses the identity operator. Suppose there exists a linear mapping F:𝒜(X)→ℒ(X) satisfying the relation F(An+2)= ∑i=1n+1Ai−1F(A2)An+1−i−∑i=1nAiF(A)An+1−i for all A∈𝒜(X) and some fixed integer n≥1. In this case F is of the form F(A)=AB1+B2A for all A∈𝒜(X) and some fixed B1,B2∈ℒ(X). In particular, F is continuous.

$$L^p$$-projections on subspaces and quotients of Banach spaces

The aim of this paper is to study \(L^p\)-projections, a notion introduced by Cunningham in 1953, on subspaces and quotients of complex Banach spaces. An \(L^p\)-projection on a Banach space X, for \(1\le p \le +\infty\), is an idempotent operator P satisfying \(\Vert f\Vert _X = \Vert ( \Vert P(f)\Vert _X, \Vert (I-P)(f)\Vert _X) \Vert _{\ell _{p}}\) for all \(f \in X\). This is an \(L^p\) version of the equality \(\Vert f\Vert ^2=\Vert Q(f)\Vert ^2 + \Vert (I-Q)(f)\Vert ^2\), valid for orthogonal projections on Hilbert spaces. We study the relationships between \(L^p\)-projections on a Banach space X and those on a subspace F, as well as relationships between \(L^p\)-projections on X and those on the quotient space X/F. All the results in this paper are true for \(1<p<+\infty\), \(p\ne 2\). The cases \(p=1,2\) or \(+\infty\) can exhibit different behavior. In this regard, we give a complete description of \(L^{\infty }\)-projections on spaces \(L^{\infty }(\Omega )\). For this, we introduce a notion of p-orthogonality for two elements x, y by requiring that \(\text {Span}(x,y)\) admits an \(L^p\)-projection separating x and y. We also introduce the notion of maximal \(L^p\)-projections for X, that is \(L^p\)-projections defined on a subspace G of X that cannot be extended to \(L^p\)-projections on larger subspaces. We prove results concerning \(L^p\)-projections and p-orthogonality of general Banach spaces or on Banach spaces with additional properties. Generalizations of some results to spaces \(L^p(\Omega ,X)\) as well as some results about \(L^q\)-projections on subspaces of \(L^p(\Omega )\) are also discussed.

Исследование некоторых классов почти периодических на бесконечности функций

The article under consideration is devoted to continuous almost periodic at infinity functions defined on the whole real axis and with their values in a complex Banach space. We consider different subspaces of functions vanishing at infinity, not necessarily tending to zero at infinity. We introduce the notions of slowly varying and almost periodic at infinity functions with respect to these subspaces. For almost periodic at infinity functions (with respect to a subspace) we give four different definitions. The first definition (approximating) is based on the approximation theorem. In the classical version, for almost periodic functions, they are represented as uniform closures of trigonometric polynomials. In our case, the Fourier coefficients are slowly varying at infinity functions. The second definition, which is an analogue of G. Bohr’s definition of an almost periodic function, is based on the concept of an ε-period. The third definition meets S. Bochner’s criterion for the almost periodicity of functions. The fourth definition is given in terms of factor space. With the help of the results of the theory of almost periodic vectors in Banach modules those four definitions are proved to be equivalent. In addition, it was proved that the introduced spaces of slowly varying and almost periodic at infinity functions with respect to different subspaces of functions vanishing at infinity coincide with the spaces of ordinary slowly varying and almost periodic at infinity functions, respectively. The feasibility of consideration of these functions is due to the fact that the solutions of some important classes of differential and difference equations are almost periodic at infinity. We consider differential equations whose right-hand side is a function vanishing at infinity and obtain necessary and sufficient conditions for their bounded solutions to be almost periodic at infinity functions. We also study an asymptotic representation of the solutions.

Bloch functions on the unit ball of a Banach space

The space of Bloch functions on bounded symmetric domains is extended by considering Bloch functions f f on the unit ball B E B_E of finite and infinite dimensional complex Banach spaces in two different ways: by extending the classical Bloch space considering the boundness of ( 1 − ‖ x ‖ 2 ) ‖ f ′ ( x ) ‖ (1-\|x\|^2) \|f’(x)\| on B E B_E and by preserving the invariance of the correspondiing seminorm when we compose with automorphisms φ \varphi of B E B_E . We study the connection between these spaces proving that they are different in general and prove that all bounded analytic functions on B E B_{E} are Bloch functions in both ways.

Hölder regularity for abstract semi-linear fractional differential equations in Banach spaces

In the present work the optimal regularity, in the sense of Hölder continuity, of linear and semi-linear abstract fractional differential equations is investigated in the framework of complex Banach spaces. This framework has been considered by the authors as the most convenient to provide a posteriori error estimates for the time discretizations of such a kind of abstract differential equations. In the spirit of the classical a posteriori error estimates, under certain assumptions, the error is bounded in terms of computable quantities, in our case measured in the norm of Hölder continuous and weighted Hölder continuous functions.

Almost periodic type functions and densities

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A note on local spectral subspace preservers

ABSTRACT Let be the Banach algebra of all bounded linear operators on a complex Banach space X. Let X T ({λ}) denote the local spectral subspace of an operator associated with a singleton , and fix a scalar . We characterize maps ϕ on which satisfy Furthermore, we give a characterization of maps ϕ on for which ϕ(0) = 0 and

STOKES-TYPE INTEGRAL EQUALITIES FOR SCALARLY ESSENTIALLY INTEGRABLE LOCALLY CONVEX VECTOR-VALUED FORMS WHICH ARE FUNCTIONS OF AN UNBOUNDED SPECTRAL OPERATOR

In this work we establish a Stokes-type integral equality for scalarly essentially integrable forms on an orientable smooth manifold with values in the locally convex linear space $\langle B(G),\sigma(B(G),\mathcal{N})\rangle$, where $G$ is a complex Banach space and $\mathcal{N}$ is a suitable linear subspace of the norm dual of $B(G)$. This result widely extends the Newton-Leibnitz-type equality stated in one of our previous articles. To obtain our equality we generalize the main result of that article, and employ the Stokes theorem for smooth locally convex vector valued forms established in a prodromic paper. Two facts are remarkable. Firstly the forms integrated involved in the equality are functions of a possibly unbounded scalar type spectral operator in $G$. Secondly these forms need not be smooth nor even continuously differentiable.

Multi-dimensional (ω, c)-almost periodic type functions and applications

Abstract In this paper, we analyze various classes of multi-dimensional (ω, c)-almost periodic type functions with values in complex Banach spaces. The main structural properties and characterizations for the introduced classes of functions are presented. We provide certain applications of our abstract theoretical results to the abstract Volterra integro-differential equations, as well.