Let $Ω$ be an irreducible bounded symmetric domain of rank $r$ in $\mathbb C^d.$ Let $\mathbb K$ be the maximal compact subgroup of the identity component $G$ of the biholomorphic automorphism group of the domain $Ω$. The group $\mathbb K$ consisting of linear transformations acts naturally on any $d$-tuple $\boldsymbol T=(T_1,\ldots, T_d)$ of commuting bounded linear operators. If the orbit of this action modulo unitary equivalence is a singleton, then we say that $\boldsymbol T$ is $\mathbb{K}$-homogeneous. In this paper, we obtain a model for all $\mathbb{K}$-homogeneous $d$-tuple $\boldsymbol{T}$ as the operators of multiplication by the coordinate functions $z_1,\ldots ,z_d$ on a reproducing kernel Hilbert space of holomorphic functions defined on $Ω$. Using this model we obtain a criterion for (i) boundedness, (ii) membership in the Cowen-Douglas class (iii) unitary equivalence and similarity of these $d$-tuples. In particular, we show that the adjoint of the $d$-tuple of multiplication by the coordinate functions on the weighted Bergman spaces are in the Cowen-Douglas class $B_1(Ω)$. For a bounded symmetric domain $Ω$ of rank $2$, an explicit description of the operator $\sum_{i=1}^d T_i^*T_i$ is given. In general, based on this formula, we make a conjecture giving the form of this operator.
We construct a large family of positive-definite kernels $K: \mathbb{D}^n\times \mathbb{D}^n \to \mbox{M} (r, \mathbb C)$, holomorphic in the first variable and anti-holomorphic in the second, that are quasi-invariant with respect to the subgroup $\mbox{Möb} \times\cdots\times \mbox{Möb}$ ($n$ times) of the bi-holomorphic automorphism group of $\mathbb{D}^n$. The adjoint of the $n$ - tuples of multiplication operators by the co-ordinate functions on the Hilbert spaces $\mathcal H_K$ determined by $K$ is then homogeneous with respect to this subgroup. We show that these $n$ - tuples are irreduc…
In this paper, we investigate the role of square functions defined for a $d$-tuple of commuting Ritt operators $(T_1,...,T_d)$ acting on a general Banach space $X$. Firstly, we prove that if the $d$-tuple admits a $H^\infty$ joint functional calculus, then it verifies various square function estimates. Then we study the converse when every $T_k$ is a $R$-Ritt operator. Under this last hypothesis, and when $X$ is a $K$-convex space, we show that square function estimates yield dilation of $(T_1,...,T_d)$ on some Bochner space $L_p(Ω;X)$ into a $d$-tuple of isomorphisms with a $C(\mathbb{T}^d)$ …
We show that if a tuple of commuting, bounded linear operators $(T_1,...,T_d) \in B(X)^d$ is both an $(m,p)$-isometry and a $(μ,\infty)$-isometry, then the tuple $(T_1^m,...,T_d^m)$ is a $(1,p)$-isometry. We further prove some additional properties of the operators $T_1,...,T_d$ and show a stronger result in the case of a commuting pair $(T_1,T_2)$.
Given a bounded operator $Q$ on a Hilbert space $\mathcal{H}$, a pair of bounded operators $(T_1, T_2)$ on $\mathcal{H}$ is said to be $Q$-commuting if one of the following holds: \[ T_1T_2=QT_2T_1 \text{ or }T_1T_2=T_2QT_1 \text{ or }T_1T_2=T_2T_1Q. \] We give an explicit construction of isometric dilations for pairs of $Q$-commuting contractions for unitary $Q$, which generalizes the isometric dilation of Ando [2] for pairs of commuting contractions. In particular, for $Q=qI_{\mathcal{H}}$, where $q$ is a complex number of modulus $1$, this gives, as a corollary, an explicit construction of …
Let $ Ω\subset \mathbb{C}^m $ be an open, connected and bounded set and $\mathcal{A}(Ω)$ be a function algebra of holomorphic functions on $Ω$. Suppose that $\mathcal{M}_q$ is the quotient Hilbert module obtained from a submodule of functions in a Hilbert module $\mathcal{M}$ vanishing to order $k$ along a smooth irreducible complex analytic set $\mathcal{Z}\subsetΩ$ of codimension at least $2$. In this article, we prove that the compression of the multiplication operators onto $\mathcal{M}_q$ is homogeneous with respect to a suitable subgroup of the automorphism group Aut$(Ω)$ of $Ω$ dependin…
Motivated by the theory of weighted shifts on directed trees and its multivariable counterpart, we address the question of identifying commutant and reflexivity of the multiplication $d$-tuple $\mathscr M_z$ on a reproducing kernel Hilbert space $\mathscr H$ of $E$-valued holomorphic functions on $Ω$, where $E$ is a separable Hilbert space and $Ω$ is a bounded domain in $\mathbb C^d$ admitting bounded approximation by polynomials. In case $E$ is a finite dimensional cyclic subspace for $\mathscr M_z$, under some natural conditions on the $B(E)$-valued kernel associated with $\mathscr H$, the c…
In this paper we give a various conditions for which the tuple $\mathcal{T} = (T_{1} , T_{2} , ... , T_{n})$ of commutative bounded linear operators on an infinite dimensional ( real , complex ) Banach space X is orbit reflexive. After we introduce the notion of weak orbit reflexive operator and we show some results.
We introduce new techniques allowing one to construct diagonals of bounded Hilbert space operators and operator tuples under "Blaschke-type" assumptions. This provides a new framework for a number of results in the literature and identifies, often large, subsets in the set of diagonals of arbitrary bounded operators (and their tuples). Moreover, our approach leads to substantial generalizations of the results due to Bourin, Herrero and Stout having assumptions of a similar nature.
In this article, we obtain two sets of results. The first set of complete results are exclusively for the case of the bi-disc while the second set of results describe in part, which of these carry over to the general case of the poly-disc: * A classification of irreducible hermitian holomorphic vector bundles over $\mathbb{D}^2$, homogeneous with respect to $\mbox{Möb}\times \mbox{Möb}$, is obtained assuming that the associated representations are \textit{multiplicity-free}. Among these the ones that give rise to an operator in the Cowen-Douglas class of $\mathbb{D}^2$ of rank $1,2$ or $3$ is …
There exist several interesting results in the literature on subnormal operator tuples having their spectral properties tied to the geometry of strictly pseudoconvex domains or to that of bounded symmetric domains in $\C^n$. We introduce a class $Ω^{(n)}$ of convex domains in $\C^n$ which, for $n \geq 2$, is distinct from the class of strictly pseudoconvex domains and the class of bounded symmetric domains and which lends itself for the application of the theories related to the abstract inner function problem and the $\bar \partial$-Neumann problem, allowing us to make a number of interesting…
Necessary and sufficient conditions for positive Toeplitz operators on the Bergman space of a minimal bounded homogeneous domain to be bounded or compact are described in terms of the Berezin transform, the averaging function and the Carleson property.
We describe the Bergman kernel of any bounded homogeneous domain in a minimal realization relating to the Bergman kernels of the Siegel disks. Taking advantage of this expression, we obtain substantial estimates of the Bergman kernel of the homogeneous domain.
Using an integral formula on a homogeneous Siegel domain, we show a necessary and sufficient condition for composition operators on the weighted Bergman space of a minimal bounded homogeneous domain to be compact. To describe the compactness of composition operators, we see a boundary behavior of the Bergman kernel.
We study the bounded and the compact weighted composition operators from the Bloch space into the weighted Banach spaces of holomorphic functions on bounded homogeneous domains, with particular attention to the unit polydisk. For bounded homogeneous domains, we characterize the bounded weighted composition operators and determine the operator norm. In addition, we provide sufficient conditions for compactness. For the unit polydisk, we completely characterize the compact weighted composition operators, as well as provide computable estimates on the operator norm.
We give conditions for boundedness of Hausdorff operators on real Hardy spaces $H^1$ over homogeneous spaces of locally compact groups with local doubling property. The special case of the hyperbolic plane is considered.
We study Toeplitz operators with respect to a commuting $n$-tuple of bounded operators which satisfies some additional conditions coming from complex geometry. Then we consider a particular such tuple on a function space. The algebra of Toeplitz operators with respect to that particular tuple becomes naturally homeomorphic to $L^\infty$ of a certain compact subset of $\mathbb C^n$. Dual Toeplitz operators are characterized. En route, we prove an extension type theorem which is not only important for studying Toeplitz operators, but also has an independent interest because dilation theorems do …
We prove a new dyadic representation theorem with applications to the $T(1)$ and $A_2$ theorems. In particular, we obtain the non-homogeneous $T(1)$ theorem under weaker kernel regularity than the earlier approaches.
We characterize functions of $d$-tuples of bounded operators on a Hilbert space that are uniformly approximable by free polynomials on balanced open sets.
We give estimates for the essential norms of a positive Toeplitz operator on the Bergman space of a minimal bounded homogeneous domain in terms of the Berezin transform or the averaging function of the symbol. Using these estimates, we also give necessary and sufficient conditions for the positive Toeplitz operators to be compact.
The aim of this note is to give the boundedness conditions for Hausdorff operators on Hardy spaces $H^{1}$ with the norm defined via $(1,q)$ atoms over homogeneous spaces of Lie groups with doubling property and to apply results we obtain to generalized Delsarte operators and to Hausdorff operators over multidimensional spheres.
In this paper, we study the multiplication operators on the Bloch space of a bounded homogeneous domain in $\mathbb{C}^n$. Specifically, we characterize the bounded and the compact multiplication operators, establish estimates on the operator norm, and determine the spectrum. We prove that the only bounded multiplication operators on the Bloch space of the polydisk are those whose symbol is constant. Furthermore, we prove that for a large class of bounded symmetric domains, the isometric multiplication operators are those whose symbol is a constant of modulus one.
We prove that the flag kernel singular integral operators of Nagel-Ricci-Stein on a homogeneous group are bounded on the Lp spaces. The gradation associated with the kernels is the natural gradation of the underlying Lie algebra. Our main tools are the Littlewood-Paley theory and a symbolic calculus combined in the spirit of Duoandikoetxea and Rubio de Francia.
Let $D$ be a bounded homogeneous domain in $\mathbb{C}^n$. In this paper, we study the bounded and the compact weighted composition operators mapping the Hardy space $H^\infty(D)$ into the Bloch space of $D$. We characterize the bounded weighted composition operators, provide operator norm estimates, and give sufficient conditions for compactness. We prove that these conditions are necessary in the case of the unit ball and the polydisk. We then show that if $D$ is a bounded symmetric domain, the bounded multiplication operators from $H^\infty(D)$ to the Bloch space of $D$ are the operators wh…
In this paper, the fractional integral operator on non-homogeneous metric measure spaces is introduced, which contains the classic fractional integral operator, fractional integral with non-doubling measures and fractional integral with fractional kernel of order $α$ and regularity $ε$ introduced by García-Cuerva and Gatto as special cases. And the $(L^{p}(μ),L^{q}(μ))$-boundedness for fractional integral operators on non-homogeneous metric measure spaces is established. From this, the $(L^{p}(μ),L^{q}(μ))$-boundedness for commutators and multilinear commutators generated by fractional integra…
Let $\x$ be the homogeneous tree of degree $q+1~(q\geq2)$. In this article, we present symbolic criteria for boundedness of pseudo-differential operators on homogeneous tree $\x$. A sufficient condition for weak type $(p, q)$ boundedness is also given. We present necessary and sufficient conditions on the symbols $σ$ such that the corresponding pseudo-differential operators $T_σ$ from $L^{p_1} (\x)$ into $L^{p_2} (\x)$ to be nuclear for $1\leq p_1, p_2<\infty$. We show that the adjoint of the nuclear operator from $L^{p_2'}(\x)$ into $L^{p_1'}(\x)$ is again a nuclear operator. Explicit formula…
In this paper, we investigate the boundedness of maximal operator and its commutators in generalized Orlicz-Morrey spaces on the spaces of homogeneous type. As an application of this boundedness, we give necessary and sufficient condition for the Adams type boundedness of fractional integral and its commutators in these spaces. We also discuss criteria for the boundedness of these operators in Orlicz spaces.
We show that the set of all possible constant diagonals of a bounded Hilbert space operator is always convex. This, in particular, answers an open question of J.-C. Bourin ($2003$). Moreover, we show that the joint numerical range of a commuting operator tuple is in general not convex, which fills a gap in the literature. We also prove that the Asplund-Ptak numerical range (which is convex for pairs of operators) is, in general, not convex for tuples of operators.
One defines a non-homogeneous space $(X, μ)$ as a metric space equipped with a non-doubling measure $μ$ so that the volume of the ball with center $x$, radius $r$ has an upper bound of the form $r^n$ for some $n> 0$. The aim of this paper is to study the boundedness of a Calderón-Zygmund operator $T$ as well as the boundedness of certain related singular integrals associated with $T$ on various function spaces on $(X, μ)$ such as the Hardy spaces, the $L^p$ spaces and the regularized BMO spaces. This article thus extends the work of X. Tolsa \cite{T1} on the non-homogeneous space $(\mathbb R^n…
A natural generalization of Krein's theorem to a pair of commuting tuples $\left(H_1^0,H_2^0\right)$ and $\left(H_1,H_2\right)$ of bounded self-adjoint operators in a separable Hilbert space $\mathcal{H}$ with $H_j-H_j^0 = V_j\in \mathcal{B}_2(\mathcal{H})$(set of all Hilbert-Schmidt operators on $\mathcal{H}$) for $j=1,2,$ leads to a Stokes-like formula under trace. A major ingredient in the proof is the finite-dimensional approximation result for commuting self-adjoint n-tuples of operators, a generalization of Weyl-von Neumann-Berg's theorem.
For graded Hilbert spaces $H$ and shift-like commuting tuples $T \in B(H)^n$, we show that each homogeneous joint invariant subspace $M$ of $T$ has finite index and is generated by its wandering subspace. Under suitable conditions on the grading $(H_k)_{k\geq 0}$ of $H$ the algebraic direct sum $\tilde{M} = \oplus_{k\geq 0} M \cap H_k$ becomes a finitely generated module over the polynomial ring $\mathbb C[z]$. We show that the wandering subspace $W_T(M)$ of $M$ is contained in $\tilde{M}$ and that each linear basis of $W_T(M)$ forms a minimal set of generators for the $\mathbb C[z]$-module $\…
For any bounded domain $Ω$ in $\mathbb C^m,$ let ${\mathrm B}_1(Ω)$ denote the Cowen-Douglas class of commuting $m$-tuples of bounded linear operators. For an $m$-tuple $\boldsymbol T$ in the Cowen-Douglas class ${\mathrm B}_1(Ω),$ let $N_{\boldsymbol T}(w)$ denote the restriction of $\boldsymbol T$ to the subspace ${\cap_{i,j=1}^m\ker(T_i-w_iI)(T_j-w_jI)}.$ This commuting $m$-tuple $N_{\boldsymbol T}(w)$ of $m+1$ dimensional operators induces a homomorphism $ρ_{_{\!N_{\boldsymbol T}(w)}}$ of the polynomial ring $P[z_1, ..., z_m],$ namely, $ρ_{_{\!N_{\boldsymbol T}(w)}}(p) = p\big (N_{\boldsym…
In this paper, we introduce homogeneous mixed Herz-Morrey spaces $M\dot{K}_{p,\vec{q}}^{α,λ}(\mathbb{R}^n)$ and show it's some properties. Firstly, the boundedness of sublinear operators, fractional type operators in homogeneous mixed Herz-Morrey spaces is investigated. In particular, the above results are still valid for Calder$\acute{o}$n-Zygmund operators and fractional maximal operators. Lastly, the boundedness of their commutators in homogeneous mixed Herz-Morrey spaces is obtained.
Let Möb denote the group of biholomorphic automorphisms of the unit disc and $(\mbox{Möb} \cdot T)$ be the orbit of a Hilbert space operator $T$ under the action of Möb. If the quotient $(\mbox{Möb} \cdot T)/\sim$, where $\sim$ is the similarity between two operators is a singleton, then the operator $T$ is said to be weakly homogeneous. In this paper, we obtain a criterion to determine if the operator $M_z$ of multiplication by the coordinate function $z$ on a reproducing kernel Hilbert space is weakly homogeneous. We use this to show that there exists a Möbius bounded weakly homogeneous oper…
We introduce a fairly general concept of functional equation for $k$-tuples of functions $f_1,\dots,f_k\colon X \to Y$ between arbitrary sets. The homomorphy equations for mappings between groups and other algebraic systems, as well as various types of functional equations and recursion formulas occurring in mathematical analysis or combinatorics, respectively, become special cases (of systems) of such equations. Assuming that $X$ is a locally compact and $Y$ is a completely regular topological space, we show that systems of such functional equations, with parameters satisfying rather a modest…
We provide two new characterizations of bounded orthogonally additive polynomials from a uniformly complete vector lattice into a convex bornological space using harmonic means and completely partitioned weighted geometric means. Our result involving completely partitioned weighted geometric means generalizes a recent theorem on bounded orthogonally additive polynomials by Z.A. Kusraeva as well as parts of related theorems by G. Buskes and the author.
It is known that all the vector bundles of the title can be obtained by holomorphic induction from representations of a certain parabolic Lie algebra on finite dimensional inner product spaces. The representations, and the induced bundles, have composition series with irreducible factors. Our first main result is the construction of an explicit differential operator intertwining the bundle with the direct sum of its factors. Next, we study Hilbert spaces of sections of these bundles. We use this to get, in particular, a full description and a similarity theorem for homogeneous $n$-tuples of op…
Results of Liflyand and collaborators on the boundedness of Hausdorff operators on the Hardy space $H^1$ over finite-dimensional real space generalized to the case of locally compact groups that are spaces of homogeneous type. Special cases and examples of compact Lie groups, homogeneous groups (in particular the Heisenberg group) and finite-dimensional spaces over division rings are considered.
We introduce a notion of a noncommutative function defined on a domain of $d$-tuples of bounded operators on an infinite dimensional Hilbert space. Inverse and implicit function theorems in this setting are established. When these operatorial noncommutative functions are suitably continuous in the strong operator topology, a noncommutative dilation-theoretic construction is used to show that the assumptions on their derivatives may be relaxed from boundedness below to merely injectivity.
We initiate the study of a class of noncommutative domains of n-tuples of bounded linear operators on a Hilbert space, which is generated by certain positivity conditions on polynomials in n noncommutative indeterminates. We obtain Fatou type results, functional calculi, and a model theory for n-tuples of operators in these domains. The main role in this study is played by a class of noncommutative Berezin transforms which is introduced in this paper. Our results extend to noncommutative varieties.
In this paper, we present the current results in the study of weighted composition operators on the Bloch space of bounded homogeneous domains in $\mathbb{C}^n$ with particular emphasis on the issues of boundedness and compactness. We also discuss the bounded and the compact weighted composition operators from the Bloch space to the Hardy space $H^\infty$.
The main features of homogeneous Cowen-Douglas operators, well-known for the unit disk, are generalized to the setting of hermitian bounded symmetric domains of arbitrary rank.
Let $(A_1,\cdots,A_n)$ and $(B_1,\cdots,B_n)$ be $n$-tuples of commuting self-adjoint operators on Hilbert space. For functions $f$ on $\R^n$ satisfying certain conditions, we obtain sharp estimates of the operator norms (or norms in operator ideals) of $f(A_1,\cdots,A_n)-f(B_1,\cdots,B_n)$ in terms of the corresponding norms of $A_j-B_j$, $1\le j\le n$. We obtain analogs of earlier results on estimates for functions of perturbed self-adjoint and normal operators. It turns out that for $n\ge3$, the methods that were used for self-adjoint and normal operators do not work. We propose a new metho…
We develop a functional calculus for $d$-tuples of non-commuting elements in a Banach algebra. The functions we apply are free analytic functions, that is nc functions that are bounded on certain polynomial polyhedra.
We show that in a bounded Gromov hyperbolic domain $Ω$ smooth functions with bounded derivatives $C^\infty(Ω)\cap W^{k,\infty}(Ω)$ are dense in the homogeneous Sobolev spaces $L^{k,p}(Ω)$.
We define and study a noncommutative Fourier transform on every homogeneous complex bounded domain. We then give an application in noncommutative differential geometry by defining noncommutative Baumslag-Solitar tori.
Let $(\cx,\,d,\,μ)$ be a metric measure space and satisfy the so-called upper doubling condition and the geometrically doubling condition. In this paper, the authors show that for the maximal Calderón-Zygmund operator associated with a singular integral whose kernel satisfies the standard size condition and the Hörmander condition, its $L^p(μ)$ boundedness with $p\in(1,\infty)$ is equivalent to its boundedness from $L^1(μ)$ into $L^{1,\infty}(μ)$. Moreover, applying this, together with a new Cotlar type inequality, the authors show that if the Calderón-Zygmund operator $T$ is bounded on $L^2(μ…
It is known that all the vector bundles of the title can be obtained by holomorphic induction from representations of a certain parabolic group on finite dimensional inner product spaces. The representations, and the induced bundles, have composition series with irreducible factors. We give a condition under which the bundle and the direct sum of its irreducible constituents are intertwined by an equivariant constant coefficient differential operator. We show that in the case of the unit ball in $\mathbb C^2$ this condition is always satisfied. As an application we show that all homogeneous pa…
