Spaces Of Holomorphic Functions Research Articles

The Jumping Operator on Invariant Subspaces in Spaces of Analytic Functions

Let D denote the Dirichlet space of holomorphic functions f in the open unit disc $$\mathbb {D}$$ with finite Dirichlet integral, $$\int _\mathbb {D}|f'|^2 dA < \infty $$ . For an $$M_z$$ -invariant subspace $$\mathcal {M}$$ of D we study the jumping operator $$P_\mathcal {M}M_z P_\mathcal {M}^{\perp }$$ from the orthogonal complement of $$\mathcal {M}$$ to $$\mathcal {M}$$ . We show that the jumping operator is in Schatten p-class for $$p > 1$$ and we obtain that for a zero-based invariant subspace $$\mathcal {M}$$ of D, the rank of the jumping operator is finite if and only if $$\mathcal {M}$$ is of finite codimension. We also prove that there are invariant subspaces of D which have infinite codimension such that the corresponding jumping operators have finite rank. Furthermore, we show that some similar results hold in the setting of the Bergman space.

Rhaly Operators on Small Weighted Hardy Spaces

We study several properties of Rhaly operators on small weighted Hardy spaces of holomorphic functions in the unit disc \(\mathbb {D}\). In particular, we obtain criteria for boundedness, compactness, Hilbert-Schmidt norm, and p-Schatten class membership of such operators. A closedness of Rhaly operator is also studied.

Precise inclusion relations among Bergman–Besov and Bloch–Lipschitz spaces and on the unit ball of

AbstractWe describe exactly and fully which of the spaces of holomorphic functions in the title are included in which others. We provide either new results or new proofs. More importantly, we construct explicit functions in each space that show our relations are strict and the best possible. Many of our inclusions turn out to be sharper than the Sobolev imbeddings.

Common reducing subspaces of several weighted shifts with operator weights

We characterize common reducing subspaces of several weighted shifts with operator weights. As applications, we study the common reducing subspaces of the multiplication operators by powers of coordinate functions on Hilbert spaces of holomorphic functions in several variables. The identification of reducing subspaces also leads to structure theorems for the commutants of von Neumann algebras generated by these multiplication operators. This general approach applies to weighted Hardy spaces, weighted Bergman spaces, Drury–Arveson spaces and Dirichlet spaces of the unit ball or polydisk uniformly.

Interpolating sequences in mean

This paper deals with interpolating sequences (zn)n for two spaces of holomorphic functions f in the unit disk D in C: those that are bounded and those that satisfy a Lipschitz condition |f(z)−f(w)|≤c|z−w|α, 0<α≤1. Given a sequence of values (wn)n in a certain target space, we look for a function f interpolating ‘in mean”, that is, with (f(z1)+⋯+f(zn))∕n=wn, n≥1. We obtain target spaces when we prescribe that the corresponding interpolating sequences be the uniformly separated ones or the union of two uniformly separated ones.

Invariant subspaces and operator model theory on noncommutative varieties

Let $${{\mathcal {Q}}}\subset {{\mathbb {C}}}\left $$ be an arbitrary set of polynomials in noncommutative indeterminates such that $$q(0)=0$$ for all $$q\in {{\mathcal {Q}}}$$ . The noncommutative variety $$\begin{aligned} {{\mathcal {V}}}_{f,{{\mathcal {Q}}}}^m({{\mathcal {H}}}):=\left\{ { X}=(X_1,\ldots , X_n)\in \mathbf{D}_f^m({{\mathcal {H}}}): \ q({ X})=0 \ \text { for all } q\in {{\mathcal {Q}}}\right\} , \end{aligned}$$ where $$\mathbf{D}_f^m({{\mathcal {H}}})$$ is a noncommutative regular domain in $$B({{\mathcal {H}}})^n$$ and $$B({{\mathcal {H}}})$$ is the algebra of bounded linear operators on a Hilbert space $${{\mathcal {H}}}$$ , admits a universal model $$B^{(m)}=(B_1^{(m)},\ldots , B_n^{(m)})$$ such that $$q(B^{(m)})=0$$ , $$q\in {{\mathcal {Q}}}$$ , acting on a model space which is a subspace of the full Fock space with n generators. In this paper, we obtain a Beurling type characterization of the joint invariant subspaces under the operators $$B_1^{(m)},\ldots , B_n^{(m)}$$ , in terms of partially isometric multi-analytic operators acting on model spaces. More generaly, a Beurling-Lax-Halmos type representation is obtained and used to parameterize the wandering subspaces of the joint invariant subspaces under $$B_1^{(m)}\otimes I_{{\mathcal {E}}},\ldots , B_n^{(m)}\otimes I_{{\mathcal {E}}}$$ , and to characterize when they are generating for the corresponding invariant subspaces. Similar results are obtained for any pure n-tuple $$(X_1,\ldots , X_n)$$ in the noncommutative variety $${{\mathcal {V}}}_{f,{{\mathcal {Q}}}}^m({{\mathcal {H}}})$$ . We characterize the elements in the noncommutative variety $${{\mathcal {V}}}_{f,{{\mathcal {Q}}}}^m({{\mathcal {H}}})$$ which admit characteristic functions, develop an operator model theory for the completely non-coisometric elements, and prove that the characteristic function is a complete unitary invariant for this class of elements. This extends the classical Sz.-Nagy–Foias functional model for completely non-unitary contractions, based on characteristic functions. Our results apply, in particular, when $${{\mathcal {Q}}}$$ consists of the noncommutative polynomials $$Z_iZ_j-Z_jZ_i$$ , $$i,j=1,\ldots , n$$ . In this case, the model space is a symmetric weighted Fock space, which is identified with a reproducing kernel Hilbert space of holomorphic functions on a Reinhardt domain in $${{\mathbb {C}}}^n$$ , and the universal model is the n-tuple $$(M_{\lambda _1},\ldots , M_{\lambda _n})$$ of multipliers by the coordinate functions.

Extension of Pettis integration: Pettis operators and their integrals

In this note, the authors discuss the concepts of a Pettis operator, by which they mean a weak $$^*$$ –weakly continuous linear operator F from a dual Banach space to an $$L_1$$ -space, and of its Pettis integral, understood simply as the dual operator $$F^*$$ of F. Applications to radial limits in weak Hardy spaces of vector-valued harmonic and holomorphic functions are provided.

Simultaneous universality

In this paper, the notion of simultaneous universality is introduced, concerning operators having orbits that simultaneously approximate any given vector. This notion is related to the well known concepts of universality and disjoint universality. Several criteria are provided, and several applications to specific operators or sequences of operators are performed, mainly in the setting of sequence spaces or spaces of holomorphic functions.

Weighted Inductive Limits of Holomorphic Functions with o-Growth Condition

We study topological properties of weighted (LB)-spaces of holomorphic functions on open sets with o-growth condition. We show that such spaces are semi-Montel if and only if they are (DFS). We also establish that, for a wide family of open sets, some important topological properties of such holomorphic spaces are always equivalent and necessary for a space to be equal algebraically to its projective hull. This means, in particular, that for such open sets these spaces behave in a very similar way as the corresponding spaces of continuous functions.

Quaternionic Hardy spaces

The theory of slice regular functions of a quaternionic vari able, as presented in [18], extends the notion of holomorphic function to the quaternionic setting. This fast growing theory is already rich of many results and has interesting applications. In this setting, the present paper is devoted to introduce and study the quaternionic counterparts of Hardy spaces of holomorphic functions of one complex variable. The basic properties of the theory of quaternionic Hardy spaces are investigated, and in particular a Poissontype representation formula, the notions of outer function , singular function and inner function are given. A quaternionic (partial) counterpart of the classical H p -factorization theorem is proved. This last result assumes a particularly interesting formulation for a large subclass of slice regular functions, where it is obtained in terms of an outer function, a singular function and a quaternionic Blaschke product. Mathematics Subject Classification (2010): 30G35, 30H10

Generic Behavior of Classes of Taylor Series Outside the Unit Disk

It is known that, generically, Taylor series of functions holomorphic in the unit disk turn out to be “maximally divergent” outside of the disk. For functions in classical Banach spaces of holomorphic functions, as for example, Hardy spaces or the disk algebra, the situation is more delicate. In this paper, it is shown that the behavior of the partial sums on sets outside the open unit disk sensitively depends on the way the sets touch the unit circle. As main tools, results on simultaneous approximation by polynomials are proved.

Isomorphism classes of weighted spaces of holomorphic functions on some subsets of complex plane

Abstract In this paper, we consider the weighted spaces of holomorphic functions on some open subsets of the complex plane and characterize the isomorphism classes of these spaces whenever our weights are warped and satisfy certain growth conditions.

Hankel and Berezin Type Operators on Weighted Besov Spaces of Holomorphic Functions on Polydisks

Let S be the space of functions of regular variation and let ω = (ω1,..., ωn), ωj ∈ S. The weighted Besov space of holomorphic functions on polydisks, denoted by B p (ω) (0 < p < +∞), is defined to be the class of all holomorphic functions f defined on the polydisk U n such that $$||f||_{{B_{P(\omega )}}}^P = \int_{{U^n}} {|Df(z){|^p}\prod\limits_{j = 1}^n {{\omega _j}{{(1 - |{z_j}{|^2})}^{P - 2}}dm{a_{2n}}(z) < \infty } } $$ , where dm2n(z) is the 2ndimensional Lebesgue measure on U n and D stands for a special fractional derivative of f.We prove some theorems concerning boundedness of the generalized little Hankel and Berezin type operators on the spaces B p (ω) and L p (ω) (the weighted L p -space).

Composition operators on weighted spaces of holomorphic functions on the upper half plane

We consider moderately growing weight functions $v$ on the upper half plane $\mathbb G$ called normal weights which include the examples $(\mathrm{Im} w)^a$, $w \in \mathbb G$, for fixed $a &gt; 0$. In contrast to the comparable, well-studied situation of normal weights on the unit disc here there are always unbounded composition operators $C_{\varphi }$ on the weighted spaces $Hv(\mathbb G)$. We characterize those holomorphic functions $\varphi \colon \mathbb G \rightarrow \mathbb G$ where the composition operator $C_{\varphi } $ is a bounded operator $Hv(\mathbb G) \rightarrow Hv(\mathbb G)$ by a simple property which depends only on $\varphi $ but not on $v$. Moreover we show that there are no compact composition operators $C_{\varphi }$ on $Hv(\mathbb G)$.

Convergence of powers of composition operators on certain spaces of holomorphic functions defined on the right half plane

This paper studies the asymptotic behaviour of the powers $$C_\varphi ^n$$ of a composition operator $$C_\varphi $$ on certain spaces of holomorphic functions defined on the right half plane $$\mathbb {C}_+$$ . It is shown that for composition operators on the Hardy spaces and the standard weighted Bergman spaces, if the inducing map $$\varphi $$ is not of parabolic type, then either the powers $$C_\varphi ^n$$ converge uniformly only to 0 or they do not converge even strongly.

From Integral Estimates of Functions to Uniform Ones. II. Sharp Versions

In the theory of functions of complex variables, exact pointwise estimates of the functions, obtained under certain integral constraints on their growth, are not common. As an example of such estimates, we can mention the pointwise estimation of the module of a function from the Fock space by its integral norm. Here we present a functional-analytic scheme for obtaining such estimates and illustrate it on the examples of the classical Fock–Bargman and Bergman–Djrbashian type spaces of holomorphic functions defined on the n-dimensional complex space, balls, polydiscs, etc.

Equivalent Bergman Spaces with Inequivalent Weights

We give a proof that every space of weighted square-integrable holomorphic functions admits an equivalent weight whose Bergman kernel has zeroes. Here the weights are equivalent in the sense that they determine the same space of holomorphic functions. Additionally, a family of radial weights in $$L^1(\mathbb {C})$$ whose associated Bergman kernels that have infinitely many zeroes is exhibited.

Outer Functions and Divergence in de Branges–Rovnyak Spaces

In most classical holomorphic function spaces on the unit disk in which the polynomials are dense, a function $f$ can be approximated in norm by its dilates $f_r(z):=f(rz)~(r<1)$, in other words, $\lim_{r\to1^-}\|f_r-f\|=0$. We construct a de Branges-Rovnyak space ${\mathcal H}(b)$ in which the polynomials are dense, and a function $f\in{\mathcal H}(b)$ such that $\lim_{r\to1^-}\|f_r\|_{{\mathcal H}(b)}=\infty$. The essential feature of our construction lies in the fact that $b$ is an outer function.

Asymptotic behavior of the powers of composition operators on Banach spaces of holomorphic functions

Asymptotic behavior of the powers of composition operators on Banach spaces of holomorphic functions

Preduals of spaces of holomorphic functions and the approximation property

Let U be a balanced open subset of a Hausdorff complex locally convex space E. In this paper we represent the predual G(U) of the space of holomorphic functions as the projective limit of the preduals of spaces of holomorphic functions that are bounded on certain subsets of U. As an application we prove that if E is a Banach space, then it has the approximation property if and only if G(U) has the approximation property. We also give a new proof of a result due to Aron and Schottenloher stating that if E is a complex Banach space with the approximation property then the space of holomorphic functions H(U), with the compact-open topology, has the approximation property.