We give an elementary proof that the $H^p$ spaces over the unit disc (or the upper half plane) are the interpolation spaces for the real method of interpolation between $H^1$ and $H^\infty$. This was originally proved by Peter Jones. The proof uses only the boundedness of the Hilbert transform and the classical factorisation of a function in $H^p$ as a product of two functions in $H^q$ and $H^r$ with $1/q+1/r=1/p$. This proof extends without any real extra difficulty to the non-commutative setting and to several Banach space valued extensions of $H^p$ spaces. In particular, this proof easily extends to the couple $H^{p_0}(\ell_{q_0}),H^{p_1}(\ell_{q_1})$, with $1\leq p_0, p_1, q_0, q_1 \leq \infty$. In that situation, we prove that the real interpolation spaces and the K-functional are induced ( up to equivalence of norms ) by the same objects for the couple $L_{p_0}(\ell_{q_0}), L_{p_1}(\ell_{q_1})$. In another direction, let us denote by $C_p$ the space of all compact operators $x$ on Hilbert space such that $tr(|x|^p) <\infty$. Let $T_p$ be the subspace of all upper triangular matrices relative to the canonical basis. If $p=\infty$, $C_p$ is just the space of all compact operators. Our proof allows us to show for instance that the space $H^p(C_p)$ (resp. $T_p$) is the interpolation space of parameter $(1/p,p)$ between $H^1(C_1)$ (resp. $T_1$) and $H^\infty(C_\infty)$ (resp. $T_ı$). We also prove a similar result for the complex interpolation method. Moreover, extending a recent result of Kaftal-Larson and Weiss, we prove that the distance to the subspace of upper triangular matrices in $C_1$ and $C_\infty$ can be essentially realized simultaneously by the same element.
It was recently shown that the theory of linear stochastic systems can be viewed as a particular case of the theory of linear systems on a certain commutative ring of power series in a countable number of variables. In the present work we study an interpolation problem in this setting. A key tool is the principle of permanence of algebraic identities.
The Carathéodory problem in the $N$-variable non-commutative Herglotz--Agler class and the Carathéodory--Fejér problem in the $N$-variable non-commutative Schur--Agler class are posed. It is shown that the Carathéodory (resp., Carathéodory--Fejér) problem has a solution if and only if the non-commutative polynomial with given operator coefficients (the data of the problem indexed by an admissible set $Λ$) takes operator values with positive semidefinite real part (resp., contractive operator values) on $N$-tuples of $Λ$-jointly nilpotent contractive $n\times n$ matrices, for all $n\in\mathbb{N…
We prove a Caratheodory-Fejer type interpolation theorem for certain matrix convex sets in $\C^d$ using the Blecher-Ruan-Sinclair characterization of abstract operator algebras. Our results generalize the work of Dmitry S. Kalyuzhnyi-Verbovetzkii for the d-dimensional non-commutative polydisc.
We continue an investigation started in a preceding paper. We discuss the classical results of Carleson connecting Carleson measures with the $\d$-equation in a slightly more abstract framework than usual. We also consider a more recent result of Peter Jones which shows the existence of a solution of the $\d$-equation, which satisfies simultaneously a good $L_ı$ estimate and a good $L_1$ estimate. This appears as a special case of our main result which can be stated as follows: Let $(Ω,\cal{A},μ)$ be any measure space. Consider a bounded operator $u:H^1\ra L_1(μ)$. Assume that on one hand $u$ …
We study a general metric constrained interpolation problem in a de Branges-Rovnyak space $\mathcal{H}(K_S)$ associated with a contractive multiplier $S$ between two Fock spaces along with its commutative counterpart, a de Branges-Rovnyak space associated with a Schur multiplier on the Drury-Arveson space of the unit ball of $\mathbb{C}^n$.
We study the stability of certain spectra under some algebraic conditions weaker than the commutativity and we generalize many known commutative perturbation results.
We reformulate, modify and extend a comparison criteria of $L^{p}$ norms obtained by Nazarov-Podkorytov and place it in the general setting of interpolation theory and majorization theory. In particular, we give norm comparison criteria for general scales of interpolation spaces, including non-commutative $L^{p}$ and Lorentz spaces. As an application, we extend the classical Ball's integral inequality, which lies at the basis of his famous result on sections of the $n-$dimensional unit cube.
In the present paper we deal with the following generalization of the sine-cosine equation \begin{equation*} \int f_1(x+y-t)+f_2(x-y+t) dμ(t)=g(x)h(y) \end{equation*} for complex valued functions $f_1$, $f_2$, $g$ and $h$ defined on a commutative topological group $G$, where $μ$ is a complex measure defined on $G$.
General results of interpolation (eg. Nevanlinna-Pick) by elements in the noncommutative analytic Toeplitz algebra $F^\infty$ (resp. noncommutative disc algebra $A_n$) with consequences to the interpolation by bounded operator-valued analytic functions in the unit ball of ${\bf C}^n$ are obtained. Non-commutative Poisson transforms are used to provide new von Neumann type inequalities. Completely isometric representations of the quotient algebra $F^\infty/J$ on Hilbert spaces, where $J$ is any $w^*$-closed, 2-sided ideal of $F^\infty$, are obtained and used to construct a $w^*$-continuous, $F^…
We give an equivalent expression for the $K$-functional associated to the pair of operator spaces $(R,C)$ formed by the rows and columns respectively. This yields a description of the real interpolation spaces for the pair $(M_n(R), M_n(C))$ (uniformly over $n$). More generally, the same result is valid when $M_n$ (or $B(\ell_2)$) is replaced by any semi-finite von Neumann algebra. We prove a version of the non-commutative Khintchine inequalities (originally due to Lust--Piquard) that is valid for the Lorentz spaces $L_{p,q}(τ)$ associated to a non-commutative measure $τ$, simultaneously for t…
We present an operator version of the Callebaut inequality involving the interpolation paths and apply it to the weighted operator geometric means. We also establish a matrix version of the Callebaut inequality and as a consequence obtain an inequality including the Hadamard product of matrices.
The paper makes the first steps into the study of extensions ("twisted sums") of noncommutative $L^p$-spaces regarded as Banach modules over the underlying von Neumann algebra $\mathcal M$. Our approach combines Kalton's description of extensions by centralizers (these are certain maps which are, in general, neither linear nor bounded) with a general principle, due to Rochberg and Weiss saying that whenever one finds a given Banach space $Y$ as an intermediate space in a (complex) interpolation scale, one automatically gets a self-extension $ 0\longrightarrow Y\longrightarrow X\longrightarrow …
We introduce the notion of a regular mapping on a non-commutative $L_p$-space associated to a hyperfinite von Neumann algebra for $1\le p\le \infty$. This is a non-commutative generalization of the notion of regular or order bounded map on a Banach lattice. This extension is based on our recent paper [P3], where we introduce and study a non-commutative version of vector valued $L_p$-spaces. In the extreme cases $p=1$ and $p=\infty$, our regular operators reduce to the completely bounded ones and the regular norm coincides with the $cb$-norm. We prove that a mapping is regular iff it is a linea…
Our starting point is a lemma due to Varopoulos. We give a different proof of a generalized form this lemma, that yields an equivalent description of the $K$-functional for the interpolation couple $(X_0,X_1)$ where $X_0=L_{p_0,\infty}(μ_1; L_q(μ_2))$ and $X_1=L_{p_1,\infty}(μ_2; L_q(μ_1))$ where $0<q<p_0,p_1\le \infty$ and $(Ω_1,μ_1), (Ω_2,μ_2)$ are arbitrary measure spaces. When $q=1$, this implies that the space $(X_0,X_1)_{θ,\infty}$ ($0<θ<1$) can be identified with a certain space of operators. We also give an extension of the Varopoulos Lemma to pairs (or finite families) of conditional …
We study real and complex interpolation of abstract Cesàro, Copson and Tandori spaces, including the description of Calderón-Lozanovski{\v ı} construction for those spaces. The results may be regarded as generalizations of interpolation for Cesàro spaces $Ces_p(I)$ in the case of real method, but they are new even for $Ces_p(I)$ in the case of complex method. Some results for more general interpolation functors are also presented. The investigations show an interesting phenomenon that there is a big difference between interpolation of Cesàro function spaces in the cases of finite and infinite …
Boehmians are quotients of sequences which are constructed by using a set of axioms. In particular, one of these axioms states that the set $S$ from which the {\it denominator} sequences are formed should be a commutative semigroup with respect to a binary operation. In this paper, we introduce a generalization of abstract Boehmian space, called $G$-Boehmian space, in which $S$ is not necessarily a commutative semigroup. Next, we provide an example of a $G$-Boehmian space and we discuss an extension of the Hartley transform on it. Finally, we compare the Hartley transform in this paper with th…
Let $\mathcal{M}$ be a semifinite von Neumann algebra. We equip the associated noncommutative $L_p$-spaces with their natural operator space structure introduced by Pisier via complex interpolation. On the other hand, for $1<p<\infty$ let $$L_{p,p}(\mathcal{M})=\big(L_{\infty}(\mathcal{M}),\,L_{1}(\mathcal{M})\big)_{\frac1p,\,p}$$ be equipped with the operator space structure via real interpolation as defined by the second named author ({\em J. Funct. Anal}. 139 (1996), 500--539). We show that $L_{p,p}(\mathcal{M})=L_{p}(\mathcal{M})$ completely isomorphically if and only if $\mathcal{M}$ is f…
We consider the closed algebra $\mathcal{A}_d$ generated by the polynomial multipliers on the Drury-Arveson space. We identify $\mathcal{A}_d^*$ as a direct sum of the preduals of the full multiplier algebra and of a commutative von Neumann algebra, and establish analogues of many classical results concerning the dual space of the ball algebra. These developments are deeply intertwined with the problem of peak interpolation for multipliers, and we generalize a theorem of Bishop-Carleson-Rudin to this setting by means of Choquet type integral representations. As a byproduct we shed some light o…
Holomorphic Interpolation of Multivariate Completely Monotone Functions
Indian Institute of Technology, Bombay（印度理工学院孟买分校）；Indian Institute of Technology Palakkad（印度理工学院帕拉卡德分校）；University of California at Santa Barbara（加利福尼亚大学圣塔芭芭拉分校）
Henkin functionals on non-commutative $\mathrm{C}^*$-algebras have recently emerged as a pivotal link between operator theory and complex function theory in several variables. Our aim in this paper is characterize these functionals through a notion of absolute continuity, inspired by a seminal theorem of Cole and Range. To do this, we recast the problem as a question in non-commutative measure theory. We develop a Glicksberg--König--Seever decomposition of the dual space of a $\mathrm{C}^*$-algebra into an absolutely continuous part and a singular part, relative to a fixed convex subset of sta…
In this article the authors study complex interpolation of Sobolev-Morrey spaces and their generalizations, Lizorkin-Triebel-Morrey spaces. Both scales are considered on bounded domains. Under certain conditions on the parameters the outcome belongs to the scale of the so-called diamond spaces.
This work explores several aspects of interpolating sequences for $\ell^p_A$, the space of analytic functions on the unit disk with $p$-summable Maclaurin coefficients. Much of this work is communicated through a Carlesonian lens. We investigate various analogues of Gramian matrices, for which we show boundedness conditions are necessary and sufficient for interpolation, including a characterization of universal interpolating sequences in terms of Riesz systems. We also discuss weak separation, giving a characterization of such sequences using a generalization of the pseudohyperbolic metric. L…
