Complex Banach Space Research Articles

Fibers and Gleason parts for the maximal ideal space of Au(Bℓp)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {A}}_u(B_{\\ell _p})$$\\end{document}

In the early nineties, R. M. Aron, B. Cole, T. W. Gamelin and W. B. Johnson initiated the study of the maximal ideal space (spectrum) of Banach algebras of holomorphic functions defined on the open unit ball of an infinite dimensional complex Banach space. Within this framework, we investigate the fibers and Gleason parts of the spectrum of the algebra of holomorphic and uniformly continuous functions on the unit ball of ℓp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _p$$\\end{document} (1≤p<∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1\\le p<\\infty $$\\end{document}). We show that the inherent geometry of these spaces provides a fundamental ingredient for our results. We prove that whenever p∈N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p\\in {\\mathbb {N}}$$\\end{document} (p≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p\\ge 2$$\\end{document}), the fiber of every z∈Bℓp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$z\\in B_{\\ell _p}$$\\end{document} contains a set of cardinal 2c\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2^{{\\mathfrak {c}}}$$\\end{document} such that any two elements of this set belong to different Gleason parts. For the case p=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p=1$$\\end{document}, we complete the known description of the fibers, showing that, for each z∈B¯ℓ1′′\\Sℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$z\\in \\overline{B}_{\\ell _1''}{\\setminus } S_{\\ell _1}$$\\end{document}, the fiber over z is not a singleton. Also, we establish that different fibers over elements in Sℓ1′′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S_{\\ell _1''}$$\\end{document} cannot share Gleason parts.

On a subclass of alpha-quasi convex mappings

In this paper, we introduce the class of α-quasi convex mappings on the unit ball in a complex Banach space, with the aim of providing a unified approach to the Fekete-Szegö problem for subclasses of normalized starlike mappings and normalized quasi-convex mappings of type B on the open unit ball B of a complex Banach space. In particular, using some restrictive assumptions, we investigate inclusion relations among the classes of alpha-quasi convex mappings, starlike mappings and quasi-convex mappings of type B.

Idempotent vector spaces and their linear transformations

Abstract This article extends topics about linear algebra and operator theoretic linear transformations on complex vector spaces to those on bicomplex spaces. For example, Definition 3 for the first time defines algebraically idempotent vector spaces, which generalizes the standard definition of a vector space and which includes bicomplex vector spaces as a special case, along with its dimension and its basis in terms of a corresponding vectorial idempotent representation. The article also shows how an n × n n\times n bicomplex matrix’s idempotent representation leads to a bicomplex Jordan form and a description of its bicomplex invariant subspace lattice diagram. Similarly, in a new way, the article rigorously defines “bicomplex Banach and Hilbert” spaces, and then it expands, for the first time, the theory of compact operators on complex Banach spaces to those on bicomplex Banach spaces. In these ways, the article indicates that the idempotent representation extends complex linear algebra and operator theory in a surprisingly generalized and straightforward way to vector space results with bicomplex and multicomplex scalars.

Operators with a non-trivial closed invariant affine subspace

We are concerned with the question of the existence of an invariant proper affine subspace for an operator A on a complex Banach space. It turns out that the presence of the number 1 in the spectrum of A or in the spectrum of its adjoint operator A∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A^*$$\\end{document} is crucial. For instance, an algebraic operator has an invariant proper affine subspace if and only if 1 is its eigenvalue. For an arbitrary operator A, we show that it has an invariant proper hyperplane if and only if 1 is an eigenvalue of A∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A^*$$\\end{document}. If A is a power bounded operator, then every invariant proper affine subspace is contained in an invariant proper hyperplane, moreover, A has a non-trivial invariant cone.

Maps preserving ascent or descent of triple Jordan product

Let X be a real or complex Banach space of dimension at least 3. We give a complete description of surjective mappings on B(X) that preserve the ascent of Jordan triple product of operators or, preserve the descent of Jordan triple product of operators.

On the coefficient inequalities for some classes of holomorphic mappings in complex Banach spaces

On the coefficient inequalities for some classes of holomorphic mappings in complex Banach spaces

Structure of strongly proximinal subspaces

In this paper we extended some results of G. Godefroy, originally proved for c0 using its special geometry, to the class of complex Banach spaces X whose dual is isometric to L1(μ). We exhibit classes of complex Banach spaces in which every strongly proximinal finite codimensional subspace is a finite dimensional perturbation of a M-ideal. We also show for this class, when the unit ball has an extreme point, if every proximinal subspace of codimension one is strongly proximinal, then the space is isometric to the k-dimensional space with the maximum norm, ℓ∞(k).

Some refinements of the Fekete–Szegö inequalities for a class of biholomorphic mappings

Let B be the unit disk U in C or the unit ball of a general complex Banach space. In this paper, we obtain a refinement of the Fekete–Szegö inequality for the class of S k + 1 0 ( B ) of mappings f for which there exist g-Loewner chains f ( z , t ) , where g ( ξ ) = 1 + ξ 1 − ξ , ξ ∈ U , such that f ( z ) = f ( z , 0 ) and z = 0 is a zero of order k + 1 of e − t f ( z , t ) − z for each t ≥ 0 . The results presented generalize the classical Fekete–Szegö inequality and the results in Hamada et al. (Fekete–Szegö problem for univalent mappings in one and higher dimensions. J Math Anal Appl. 2022;516:126526).

The Fekete and Szegő Problem for a Class of Holomorphic Mappings Associated with Starlike Mappings on the Unit Balls of Complex Banach Spaces

The Fekete and Szegő Problem for a Class of Holomorphic Mappings Associated with Starlike Mappings on the Unit Balls of Complex Banach Spaces

Linearization of holomorphic Lipschitz functions

AbstractLet and be complex Banach spaces with denoting the open unit ball of . This paper studies various aspects of the holomorphic Lipschitz space , endowed with the Lipschitz norm. This space consists of the functions in the intersection of the sets of Lipschitz mappings and of bounded holomorphic mappings, from to . Thanks to the Dixmier–Ng theorem, is indeed a dual space, whose predual shares linearization properties with both the Lipschitz‐free space and Dineen–Mujica predual of . We explore the similarities and differences between these spaces, and combine techniques to study the properties of the space of holomorphic Lipschitz functions. In particular, we get that contains a 1‐complemented subspace isometric to and that has the (metric) approximation property whenever has it. We also analyze when is a subspace of , and we obtain an analog of Godefroy's characterization of functionals with a unique norm preserving extension in the holomorphic Lipschitz context.

ESTABLISH OF THE JENSEN TYPE ( Γ_1, Γ_2 )-FUNCTIONAL INEQUATITIES BASED ON JENSEN TYPE FUNCTIONAL EQUATION WITH 3k-VARIABLES IN COMPLEX BANACH SPACE

In this paper, I work on expanding the Jensen $(\Gamma_{1},\Gamma_{2})$-function inequalities by relying on the general Jensen functional equation with 3k-variables on the complex Banach space. That's the main result in this.

P-Summing Bloch mappings on the complex unit disc

The notion of p-summing Bloch mapping from the complex unit open disc D\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {D}$$\\end{document} into a complex Banach space X is introduced for any 1≤p≤∞.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1\\le p\\le \\infty .$$\\end{document} It is shown that the linear space of such mappings, equipped with a natural seminorm πpB,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pi ^{\\mathcal {B}}_p,$$\\end{document} is Möbius-invariant. Moreover, its subspace consisting of all those mappings which preserve the zero is an injective Banach ideal of normalized Bloch mappings. Bloch versions of the Pietsch’s domination/factorization Theorem and the Maurey’s extrapolation Theorem are presented. We also introduce the spaces of X-valued Bloch molecules on D\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {D}$$\\end{document} and identify the spaces of normalized p-summing Bloch mappings from D\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {D}$$\\end{document} into X∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X^*$$\\end{document} under the norm πpB\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pi ^{\\mathcal {B}}_p$$\\end{document} with the duals of such spaces of molecules under the Bloch version of the p∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p^*$$\\end{document}-Chevet–Saphar tensor norms dp∗.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d_{p^*}.$$\\end{document}

Recurrent Subspaces in Banach Spaces

Abstract We study the spaceability of the set of recurrent vectors $\text{Rec}(T)$ for an operator $T:X\longrightarrow X$ on a Banach space $X$. In particular, we find sufficient conditions for a quasi-rigid operator to have a recurrent subspace; when $X$ is a complex Banach space, we show that having a recurrent subspace is equivalent to the fact that the essential spectrum of the operator intersects the closed unit disk, and we extend the previous result to the real case. As a consequence, we obtain that a weakly-mixing operator on a real or complex separable Banach space has a hypercyclic subspace if and only if it has a recurrent subspace. The results exposed exhibit a symmetry between the hypercyclic and recurrent spaceability theories showing that, at least for the spaceable property, hypercyclicity and recurrence can be treated as equals.

Well‐posedness of degenerate fractional differential equations with finite delay in complex Banach spaces

AbstractWe study the well‐posedness of the degenerate fractional differential equations with finite delay: on Lebesgue–Bochner spaces and periodic Besov spaces , where and are closed linear operators in a complex Banach space satisfying , and are fixed, when , and the delay operator is a bounded linear operator from (resp. ) into . Using known operator‐valued Fourier multiplier theorems on and , we completely characterize the ‐well‐posedness and the ‐well‐posedness of above equations. We also give concrete examples that our abstract results may be applied.

ISOMETRIES AND HERMITIAN OPERATORS ON SPACES OF VECTOR-VALUED LIPSCHITZ MAPS

Abstract We study hermitian operators and isometries on spaces of vector-valued Lipschitz maps with the sum norm. There are two main theorems in this paper. Firstly, we prove that every hermitian operator on $\operatorname {Lip}(X,E)$ , where E is a complex Banach space, is a generalized composition operator. Secondly, we give a complete description of unital surjective complex linear isometries on $\operatorname {Lip}(X,\mathcal {A})$ , where $\mathcal {A}$ is a unital factor $C^{*}$ -algebra. These results improve previous results stated by the author.

Algebras of weakly symmetric functions on spaces of Lebesgue measurable functions

In this work, we investigate algebras of block-symmetric and weakly symmetric polynomials and analytic functions on complex Banach spaces of Lebesgue measurable functions, for which the $p$th power of the absolute value is Lebesgue integrable, where $p\in[1,+\infty),$ and Lebesgue measurable essentially bounded functions on $[0,1].$ We construct generating systems of algebras of all weakly symmetric continuous complex-valued polynomials on these spaces. Also we establish conditions under which sets of weakly symmetric analytic functions are algebras.

The p-Bohr radius for vector-valued holomorphic and pluriharmonic functions

Abstract We study a “p-powered” version K n p ⁢ ( F ⁢ ( R ) ) {K_{n}^{p}(F(R))} of the well-known Bohr radius problem for the family F ⁢ ( R ) {F(R)} of holomorphic functions f : R → X {f:R\to X} satisfying ∥ f ∥ &lt; ∞ {\lVert f\rVert&lt;\infty} , where ∥ ⋅ ∥ {\lVert\,\cdot\,\rVert} is a norm in the function space F ⁢ ( R ) {F(R)} , R ⊂ ℂ n {R\subset{\mathbb{C}}^{n}} is a complete Reinhardt domain, and X is a complex Banach space. For all p &gt; 0 {p&gt;0} , we describe in full detail the asymptotic behavior of K n p ⁢ ( F ⁢ ( R ) ) {K_{n}^{p}(F(R))} , where F ⁢ ( R ) {F(R)} is: (a) the Hardy space of X-valued holomorphic functions defined in the open unit polydisk 𝔻 n {{\mathbb{D}}^{n}} ; and (b) the space of bounded X-valued holomorphic or complex-valued pluriharmonic functions defined in the open unit ball B ⁢ ( l t n ) {B(l_{t}^{n})} of the Minkowski space l t n {l_{t}^{n}} . We give an alternative definition of the optimal cotype for a complex Banach space X in the light of these results. In addition, the best possible versions of two theorems from [C. Bénéteau, A. Dahlner and D. Khavinson, Remarks on the Bohr phenomenon, Comput. Methods Funct. Theory 4 2004, 1, 1–19] and [S. Chen and H. Hamada, Some sharp Schwarz–Pick type estimates and their applications of harmonic and pluriharmonic functions, J. Funct. Anal. 282 2022, 1, Paper No. 109254] have been obtained as specific instances of our results.

Linear maps preserving equivalence or asymptotic equivalence on Banach space

Abstract Let X X be a complex Banach space with dimension at least two and B ( X ) B\left(X) the algebra of all bounded linear operators on X X . We show that a bijective linear map Φ \Phi preserves asymptotic equivalence if and only if it preserves equivalence, and in turn, if and only if there exist invertible bounded linear operators T T and S S such that either Φ ( A ) = T A S \Phi \left(A)=TAS or Φ ( A ) = T A * S \Phi \left(A)=T{A}^{* }S for all A ∈ B ( X ) A\in B\left(X) .

Coefficient Problems of Quasi-Convex Mappings of Type B on the Unit Ball in Complex Banach Spaces

ABSTRACT In this paper, the sharp solutions of Fekete-Szegö problems are provided for class of quasi-convex mappings f 1 of type B and class of quasi-convex mappings f 2 of type B and order α defined on the unit ball in a complex Banach space, respectively, where x = 0 is a zero of order k + 1 of fi (x) − x (i = 1, 2). Compare with some recent works, our main theorems hold without additional restrictive conditions. Also, the proof of our main theorems are more simple than those given in the previous results.

A new version of the Gelfand-Hille theorem

Let X be a complex Banach space and A∈L(X) with σ(A)={1}. We prove that for a vector x∈X, if ‖(Ak+A−k)x‖=O(kN) as k→+∞ for some positive integer N, then (A−I)N+1x=0 when N is even and (A−I)N+2x=0 when N is odd. This could be seemed as a new version of the Gelfand-Hille theorem. As a corollary, we also obtain that for a quasinilpotent operator Q∈L(X) and a vector x∈X, if ‖cos⁡(kQ)x‖=O(kN) as k→+∞ for some positive integer N, then QN+1x=0 when N is even and QN+2x=0 when N is odd. Moreover, if A satisfies the uniqueness property for the local resolvent, the condition σ(A)={1} could be replaced by and σx(A)={1}, which is a more local condition.