T. Constantinescu, J. L. Johnson
In this paper we formulate and solve Nevanlinna-Pick and Carathéodory type problems for tensor algebras with data given on the N-dimensional operator unit ball of a Hilbert space. We develop an approach based on the displacement structure theory.
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Noncommutative commutant lifting, interpolation, and shift-invariant subspaces
本文在非交换Drury-Arveson空间中建立Sarason型交换子提升定理，解决非交换Nevanlinna-Pick插值问题，并给出移位不变子空间的Beurling-Lax-Halmos型刻画。
The Schur-Agler class consists of functions over a domain satisfying an appropriate von Neumann inequality. Originally defined over the polydisk, the idea has been extended to general domains in multivariable complex Euclidean space with matrix polynomial defining function as well as to certain multivariable noncommutative-operator domains with a noncommutative linear-pencil defining function. Still more recently there has emerged a free noncommutative function theory (functions of noncommuting matrix variables respecting direct sums and similaritytransformations). The purpose of the present p…
We provide an effective single-matrix criterion, in terms of what we call the elementary Pick matrix, for the solvability of the noncommutative Nevanlinna-Pick interpolation problem in the row ball, and provide some applications. In particular we show that the so-called "column-row property" fails for the free semigroup algebras, in stark contrast to the analogous commutative case. Additional applications of the elementary Pick matrix include a local dilation theorem for matrix row contractions and interpolating sequences in the noncommutative setting. Finally we present some numerical results…
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 studied complex interpolation noncommutative Hardy space associated with semi-finite von Neumann algebra and extend Pisier's interpolation theorem for this case.
A noncommutative Bishop peak interpolation-set theorem
本文证明了Bishop峰值插值集定理的非交换版本，扩展了该经典插值定理在非交换框架下的适用性。
We prove that atomic decomposition for the Hardy spaces h_1 and H_1 is valid for noncommutative martingales. We also establish that the conditioned Hardy spaces of noncommutative martingales h_p and bmo form interpolation scales with respect to both complex and real interpolations.
The class of Schur-Agler functions over a domain ${\mathcal D} \subset {\mathbb C}^{d}$ is defined as the class of holomorphic operator-valued functions on ${\mathcal D}$ for which a certain von Neumann inequality is satisfied when a commuting tuple of operators satisfying a certain polynomial norm inequality is plugged in for the variables. Such functions are alternatively characterized as those having a linear-fractional presentation which identifies them as transfer functions of a certain type of conservative structured multidimensional linear system. There now has been introduced a noncomm…
This paper is devoted to the study of noncommutative weak Orlicz spaces and martingale inequalities. Marcinkiewicz interpolation theorem is extended to include noncommutative weak Orlicz spaces as interpolation classes. In particular, we prove the weak type $Φ$-moment Burkholder-Gundy inequality for noncommutative martingales through establishing a weak type $Φ$-moment noncommutative Khintchine's inequality for Rademacher's random variables.
We prove a Boyd-type interpolation result for noncommutative maximal operators of restricted weak type. Our result positively answers an open question posed recently by Bekjan, Chen and Osekowski. As a special case, we find a restricted weak type version of the noncommutative Marcinkiewicz interpolation theorem, due to Junge and Xu, with interpolation constant of optimal order.
We present a new, elementary proof of Boyd's interpolation theorem. Our approach naturally yields a noncommutative version of this result and even allows for the interpolation of certain operators on l^1-valued noncommutative symmetric spaces. By duality we may interpolate several well-known noncommutative maximal inequalities. In particular we obtain a version of Doob's maximal inequality and the dual Doob inequality for noncommutative symmetric spaces. We apply our results to prove the Burkholder-Davis-Gundy and Burkholder-Rosenthal inequalities for noncommutative martingales in these spaces…
In this paper, we initiate the study of sub-pluriharmonic curves in Cuntz-Toeplitz algebras and free pluriharmonic majorants on noncommutative balls. We are lead to a characterization of the noncommutative Hardy space $H^2_{\bf ball}$ in terms of free pluriharmonic majorants, and to a Schur type description of the unit ball of $H^2_{\bf ball}$. These results are used to solve a multivariable commutant lifting problem and provide a description of all solutions.
In this short note, we answer two questions about Gurariy spaces asked in the literature in the affirmative. We also prove the analogue of one of the results for the noncommutative Guariy space.
Let $0<p<q\leq\infty$ and $α\in (0,\infty]$. We give a characterization of quasi-Banach interpolation spaces for the couple $(L_p(0,α),L_q(0,α))$ in terms of two monotonicity properties, extending known results which mainly dealt with Banach spaces. This enables us to recover recent results of Cwikel and Nilsson on sequence spaces and to solve a conjecture of Levitina, Sukochev and Zanin in the setting of function spaces. We apply the results obtained to characterize symmetric spaces in which the standard forms of the noncommutative Khintchine inequalities hold.
Let $\mathcal{M}$ be a ($σ$-finite) von Neumann algebra associated with a normal faithful state $ϕ.$ We prove a complex interpolation result for a couple of two (quasi) Haagerup noncommutative $L_p$-spaces $L_{p_0} (\mathcal{M}, ϕ)$ and $L_{p_1} (\mathcal{M}, ϕ), 0< p_0 < p_1\leq \infty,$ which has further applications to the sandwiched $p$-Rényi divergence.
We prove noncommutative Khintchine inequalities for all interpolation spaces between $L_p$ and $L_2$ with $p<2$. In particular, it follows that Khintchine inequalities hold in $L_{1,\infty}$. Using a similar method, we find a new deterministic equivalent for the $RC$-norm in all interpolation spaces between $L_p$-spaces which unifies the cases $p > 2$ and $p < 2$. It produces a new proof of Khintchine inequalities for $p<1$ for free variables. To complete the picture, we exhibit counter-examples which show that neither of the usual closed formulas for Khintchine inequalities can work in $L_{2,…
Peak interpolation is concerned with a foundational kind of mathematical task: building functions in a fixed algebra $A$ which have prescribed values or behaviour on a fixed closed subset (or on several disjoint subsets). In this paper we do the same but now $A$ is an algebra of operators on a Hilbert space. We briefly survey this {\em noncommutative peak interpolation}, which we have studied with coauthors in a long series of papers, and whose basic theory now appears to be approaching its culmination. This program developed from, and is based partly on, theorems of Hay and Read whose proofs …
A note on $L^p$-$L^q$ boundedness of Fourier multipliers on noncommutative spaces
该研究针对非交换空间上的Fourier乘子展开，给出了半有限von Neumann代数关联的一般非交换空间上Fourier乘子$L^p$-$L^q$估计的简洁证明，覆盖局部紧幺模群上的相关情形。
Let $\mathcal{M}$ be a semifinite von Nemann algebra equipped with an increasing filtration $(\mathcal{M}_n)_{n\geq 1}$ of (semifinite) von Neumann subalgebras of $\mathcal{M}$. For $0<p <\infty$, let $\mathsf{h}_p^c(\mathcal{M})$ denote the noncommutative column conditioned martingale Hardy space and $\bmo^c(\M)$ denote the column \lq\lq little\rq\rq \ martingale BMO space associated with the filtration $(\mathcal{M}_n)_{n\geq 1}$. We prove the following real interpolation identity: if $0<p <\infty$ and $0<θ<1$, then for $1/r=(1-θ)/p$, \[ \big(\mathsf{h}_p^c(\mathcal{M}), \bmo^c(\mathcal{M})\…
Let $\mathcal{M}$ be a semifinite von Neumann algebra equipped with a semifinite normal faithful trace $τ$. Let $d$ be an injective positive measurable operator with respect to $(\mathcal{M}, τ)$ such that $d^{-1}$ is also measurable. Define $$L_p(d)={x\in L_0(\mathcal{M}) : dx+xd\in L_p(\mathcal{M})}\quad{and}\quad \|x\|_{L_p(d)}=\|dx+xd\|_p .$$ We show that for $1\le p_0<p_1\le\8$, $0<θ<1$ and $α_0\ge0, α_1\ge0$ the interpolation equality $$(L_{p_0}(d^{α_0}), L_{p_1}(d^{α_1}))_θ =L_{p}(d^α)$$ holds with equivalent norms, where $\frac1p=\frac{1-θ}{p_0}+\fracθ{p_1}$ and $α=(1-θ)α_0+θα_1$.
Let $\mathcal{M}$ be a semifinite von Nemann algebra equipped with an increasing filtration $(\mathcal{M}_n)_{n\geq 1}$ of (semifinite) von Neumann subalgebras of $\M$. For $0<p \leq\infty$, let $\h_p^c(\mathcal{M})$ denote the noncommutative column conditioned martingale Hardy space associated with the filtration $(\mathcal{M}_n)_{n\geq 1}$ and the index $p$. We prove that for $0<p<\infty$, the compatible couple $\big(\h_p^c(\mathcal{M}), \h_\infty^c(\mathcal{M})\big)$ is $K$-closed in the couple $\big(L_p(\mathcal{N}), L_\infty(\mathcal{N}) \big)$ for an appropriate amplified semifinite von …
K-closedness results in noncommutative Lebesgue spaces with filtrations
本文在带滤过的非交换Lebesgue空间实插值中建立广义K-闭性结果，并借此解决Randrianantoanina提出的非交换鞅Hardy空间K-闭性问题，方法上推广了Bourgain的经典Hardy空间插值技术。
A note on Pisier's method in interpolation of abstract Hardy spaces
本文在更一般背景下提出并证明了Pisier方法的抽象版本，应用于非交换鞅变换研究。
Known or essentially known results about duals of interpolation spaces are presented, taking a point of view sometimes slightly different from the usual one. Particular emphasis is placed on Alberto Calderon's theorem describing the duals of his complex interpolation spaces [A_0,A_1]_θ. The pace is slow, since these notes are intended for graduate students who have just begun to study interpolation spaces. This second version corrects some small misprints. It also draws attention to a convenient norming subspace of the dual of a complex interpolation space, and to the slight difference between…
Free analysis is a quantization of the usual function theory much like operator space theory is a quantization of classical functional analysis. Basic objects of free analysis are noncommutative functions. These are maps on tuples of matrices of all sizes that preserve direct sums and similarities. This paper investigates the local theory of noncommutative functions. The first main result shows that for a scalar point $Y$, the ring $O_Y$ of uniformly analytic noncommutative germs about $Y$ is an integral domain and admits a universal skew field of fractions, whose elements are called meromorph…
Realization theory for operator colligations on Pontryagin spaces is used to study interpolation and factorization in generalized Schur classes. Several criteria are derived which imply that a given function is almost the restriction of a generalized Schur function. The role of realization theory in coefficient problems is also discussed.
A Note on Complex Interpolation of Quasi-Banach Function Spaces
本文改进了 Kalton-Mitrea 关于拟巴拿赫函数空间复插值的刻画，去除了一个可分性条件，并推广了 Wolff 迭代定理。
These lecture notes were written during a mini-course on noncommutative Lp-spaces at the Basque Center of Applied Mathematics. It starts presenting the theory of weights and traces in von Neumann algebra, followed by the theory of noncommutative measure. After that, it presents a construction of these spaces in all detail and give some important results e.g. the noncommutative Radon-Nikodym theorem.
Gagliardo-Nirenberg interpolation inequality for symmetric spaces on Noncommutative torus
本文针对非交换环面上的对称算子空间，在Cesàro算子有界的条件下，建立并证明了Gagliardo-Nirenberg插值不等式，并提出了一种有别于以往设定的全新证明方法。
An indefinite generalization of Nudel'man's problem is used in a systematic approach to interpolation theorems for generalized Schur and Nevanlinna functions with interior and boundary data. Besides results on existence criteria for Pick-Nevanlinna and Caratheodory-Fejer interpolation, the method yealds new results on generalized interpolation in the sense of Sarason and boundary interpolation, including properties of the finite Hilbert transform relative to weights. The main theorem appeals to the Ball and Helton almost-commutant lifting theorem to provide criteria for the existence of a solu…
Orlicz Space Interpolation and Its Applications to Operator Convolution
本文研究了非交换Orlicz空间的插值理论，并应用该理论得到Weyl伪微分符号和Werner函数-算子卷积的Young型卷积估计。
A note concerning fundamental functions of interpolation associated to the inverse multiquadric $(α^2+x^2)^{-k}$
本文针对逆多重二次函数 $(α^2+x^2)^{-k}$ 的散乱平移，构造并研究了相应的基本插值函数。
We present the fractional perimeter as a set-function interpolation between the Lebesgue measure and the perimeter in the sense of De Giorgi. Our motivation comes from a new fractional Boxing inequality that relates the fractional perimeter and the Hausdorff content and implies several known inequalities involving the Gagliardo seminorm of the Sobolev spaces $W^{α, 1}$ of order $0 < α< 1$.
P. Jones' interpolation theorem for noncommutative martingale Hardy spaces II
本文证明半有限 von Neumann 代数上非交换列鞅 Hardy 空间在 $\mathcal{H}_1^c$ 与 $\mathcal{H}_\infty^c$ 之间的实插值恒等式，并推广至 Orlicz-Hardy 空间。
Notes on Bernstein spaces, sampling, Boas interpolation formulas and their extensions to Banach spaces
本文综述了调和分析中经典结果及其在巴纳赫空间中的扩展，涵盖伯恩斯坦空间、香农采样和博斯插值公式等核心内容。
In this paper we obtain a noncommutative multivariable analogue of the classical Nevanlinna-Pick interpolation problem for analytic functions with positive real parts on the open unit disc. As consequences, we deduce some results concerning operator-valued analytic interpolation on the unit ball on C^n.
We prove the noncommutative Davis decomposition for the column Hardy space $\H_p^c$ for all $0<p\leq 1$. A new feature of our Davis decomposition is a simultaneous control of $\H_1^c$ and $\H_q^c$ norms for any noncommutative martingale in $\H_1^c \cap \H_q^c$ when $q\geq 2$. As applications, we show that the Burkholder/Rosenthal inequality holds for bounded martingales in a noncommutative symmetric space associated with a function space $E$ that is either an interpolation of the couple $(L_p, L_2)$ for some $1<p<2$ or is an interpolation of the couple $(L_2, L_q)$ for some $2<q<\infty$. We al…
In this paper, we complete the study of mapping properties for a family of operators evaluating the difference between differentiation operators and conditional expectations acting on noncommutative $L_{p}$-spaces. To be more precise, we establish the weak type $(1,1)$ and $(L_{\infty},\mathrm{BMO})$ estimates of this difference. Consequently, in conjunction with interpolation and duality, we obtain all strong type $(p,p)$ estimates. This allows us to obtain a quick application to noncommutative differential transforms for averaging operators.
In this short note, we merge the areas of hypercomplex algebras with that of fractal interpolation and approximation. The outcome is a new holistic methodology that allows the modelling of phenomena exhibiting a complex self-referential geometry and which require for their description an underlying algebraic structure.
Suppose that the linear operator $T$ maps $X_0$ compactly to $Y_0$ and also maps $X_1$ boundedly to $Y_1$. We deal once again with the 51 year old question of whether $T$ also always maps the complex interpolation space $[X_0,X_1]_θ$ compactly to $[Y_0,Y_1]_θ$. This is a short preliminary version of our promised technical sequel to our earlier paper arXiv:1410.4527 on this topic. It contains the following two small new partial results: (i) The answer to the above question is yes, in the particular case where $Y_0$ is a UMD-space. (ii) The answer to the above question is yes for given spaces $X…
Denote by $M_n$ the algebra of $n\times n$ matrices. We consider the dyadic paraproducts $π_b$ associated with $M_n$ valued functions $b$, and show that the $L^\infty (M_n)$ norm of $b$ does not dominate $||π_b||_{L^2(\ell _n^2)\to L^2(\ell_n^2)}$ uniformly over $n$. We also consider paraproducts associated with noncommutative martingales and prove that their boundedness on bounded noncommutative $L^p-$% martingale spaces implies their boundedness on bounded noncommutative $L^q-$% martingale spaces for all $1<p<q<\infty $.
Notes on noncommutative ergodic theorems
本文证明了半有限von Neumann代数中算子网在多种收敛下的完备性，并将逐点Cauchy性质扩展至完全对称空间，为非交换遍历理论提供了应用基础。
Null projections and noncommutative function theory in operator algebras
本文研究算子代数中关于子代数为零的投影，将其与峰值投影和经典零集类比，通过推广Varopoulos定理刻画零投影条件，并应用于峰值插值及非交换函数论多个主题。
In this paper, we investigate the boundedness of a square function from ergodic theory on noncommutative $L_{p}$-spaces. The main result is a weak $(1,1)$ type estimate of this square function. We also show the $(L_{\infty},\mathrm{BMO})$ estimate, and thus strong $(L_{p},L_{p})$ estimate by interpolation. The main novel difficulty lies in the fact that the kernel of this square function does not enjoy any regularity, which is crucial in showing such endpoint estimates for standard noncommutative Calderón-Zygmund singular integral operators.
A Short Note on the Comparison of Interpolation Widths, Entropy Numbers, and Kolmogorov Widths
本文比较了紧算子从希尔伯特空间到巴拿赫空间的科莫戈洛夫和熵数，并将其应用于再生核希尔伯特空间与L_∞(μ)之间的嵌入，给出了插值宽度与科莫戈洛夫n-宽度之间n^{1/2}级差距的充分条件，并证明在多维索伯列夫情况下，科莫戈洛夫与近似宽度之间确实存在此差距。
Schatten classes on noncommutative tori: Kernel conditions
本文研究了量子环面上积分算子的Schatten类隶属问题，通过引入非交换积分核准则并结合非交换Schwartz核定理，建立了判定有界算子Schatten类性质的具体测试方法。
These notes contain a presentation of the noncommutative generalization of the classical moment problem introduced in [10] and [12]. They also contain a short summary of the classical moment problem in infinite dimension.
Fractional Sobolev spaces via interpolation, and applications to mixed local-nonlocal operators
本文介绍分数阶Sobolev空间与插值理论的关联，给出其作为Lebesgue与整数阶Sobolev空间间实/复插值空间的等价刻画，并用插值理论证明混合局部-非局部算子的谱定理，为早期研究者提供该领域简洁入门内容。
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