Operator factorization of range space relations

Factorizations of EP Banach space operators and EP Banach algebra elements

EP Banach space operators and EP Banach algebra elements are characterized using different kinds of factorizations. The results obtained generalize well-known characterizations of EP matrices, EP Hilbert space operators and EP $C^*$-algebra elements. Furthermore, new results that hold in these contexts are presented.

Factorization theory for a class of Toeplitz + Hankel operators

In this paper we study operators of the form $M(ϕ)=T(ϕ)+H(ϕ)$ where $T(ϕ)$ and $H(ϕ)$ are the Toeplitz and Hankel operators acting on $H^p(\T)$ with generating function $ϕ\in L^\iy(\T)$. It turns out that $M(ϕ)$ is invertible if and only if the function $ϕ$ admits a certain kind of generalized factorization.

Factorizations of weighted EP Banach space operators and Banach algebra elements

Weighted EP Banach space operators and Banach algebra elements are characterized using different kinds of factorizations. The results presented extend well-known characterizations of (weighted) EP matrices, (weighted) EP Hilbert space operators and (weighted) EP $C^*$-algebra elements.

Factorization of completely bounded weakly compact operators

We prove that in the setting of operator spaces the result of Davis, Figiel, Johnson and Pelczynski on factoring weakly compact operators holds accordingly. Though not related directly to the main theorem we add a remark on the description of weakly compact subsets in the dual of noncommutative vector valued ${\rm L}_1$.

反线性算子与 $C$-正规算子的分解

Factorization of anti-linear and $C$-normal operators

本文研究反线性及 $C$-正规算子，利用 Douglas 定理建立反线性算子的极分解与值域包含关系，并给出 $C$-正规算子分解及共轭正规算子的 Cartesian 分解，推广已有结果。

Factorizations of Linear Relations

Given two linear relations $A$ and $B$ we characterize the existence of a linear relation (operator) $C$ such that $A\subseteq BC$, respectively $A\subseteq CB.$ These factorizations extend and improve well-known results by R.G. Douglas and Z. Sebestyén.

On factorization of operators between Banach spaces

A finite-dimensional analogue of the known Gordon-Lewis constant of a Banach space X is introduced; in its definition are used only finite rank operators. It is shown that there exist Banach spaces such that the standard Gordon-Lewis constant of X is finite and its finite-dimensional analogue GLfin(X) is infinite. Moreover, for any Banach space X of cotype 2 its finite-dimensional constant GLfin(X) is finite too.

Linear Factorization of Hypercyclic Functions for Differential Operators

On the Fréchet space of entire functions $H(\mathbb{C})$, we show that every nonscalar continuous linear operator $L:H(\mathbb{C})\to H(\mathbb{C})$ which commutes with differentiation has a hypercyclic vector $f(z)$ in the form of the infinite product of linear polynomials: \[ f(z) = \prod_{j=1}^\infty \, \left( 1-\frac{z}{a_j}\right), \] where each $a_j$ is a nonzero complex number.

Factorization Theorem through a Dunford-Pettis $p$-convergent operator

In this article, we introduce the notion of $p$-$(DPL)$ sets.\ Also, a factorization result for differentiable mappings through Dunford-Pettis $p$-convergent operators is investigated.\ Namely, if $ X ,Y $ are real Banach spaces and $U$ is an open convex subset of $X,$ then we obtain that, given a differentiable mapping $f: U\rightarrow Y$ its derivative $f^{\prime}$ takes $U$-bounded sets into $p$-$(DPL)$ sets if and only if it happens $f=g\circ S,$ where $S$ is a Dunford-Pettis $p$-convergent operator from $X$ into a suitable Banach space $Z$ and $g:S(U)\rightarrow Y$ is a Gâteaux differenti…

线性关系的幂等因子分解

Factorizations of linear relations by idempotents

该研究针对可分解为幂等关系乘积的线性关系类别展开，给出其若干刻画，将算子分解相关已知结果推广至更一般的线性关系场景。

Cascade Connections of Linear Systems and Factorizations of Holomorphic Operator Functions Around a Multiple Zero in Several Variables

We show that the factorization problem $θ(z)=θ_2(z)θ_1(z)$ is solvable in the class of Hilbert space operator-valued functions holomorphic on some neighbourhood of $z=0$ in $\nspace{C}{N}$ and having a zero at $z=0$ (here $θ(z)$ has a multiple zero at $z=0$). Such a factorization problem becomes more complicated if we demand for $θ(z), θ_1(z)$ and $θ_2(z)$ to be Agler--Schur-class functions on the polydisk $\nspace{D}{N}$ and for the factorization identity to hold in $\nspace{D}{N}$. In this case we reduce it to the problem on the existence of a cascade decomposition for certain multiparametri…

Loops in SU(2) and Factorization, II

In the prequel to this paper, we proved that for a $SU(2,\mathbb C)$ valued loop having the critical degree of smoothness (one half of a derivative in the $L^2$ Sobolev sense), the following statements are equivalent: (1) the Toeplitz and shifted Toeplitz operators associated to the loop are invertible, (2) the loop has a unique triangular factorization, and (3) the loop has a unique root subgroup factorization. This hinges on some Plancherel-esque formulas for determinants of Toeplitz operators. The main point of this report is is to outline a generalization of this result to loops of vanishi…

An operator algebraic proof of Agler's factorization theorem

We give a short direct proof of Agler's factorization theorem that uses the abstract characterization of operator algebras. the key ingredient of this proof is an operator algebra factorization theorem. Our proof provides some additional information about these factorizations in the case of polynomials.

算子在Kreĭn空间上的自共轭算子的赫米特指数与分解

Hermitian indices and factorization of selfadjoint operators on a Kreĭn space

本文研究了自共轭算子在Kreĭn空间上的赫米特指数与分解，证明了在不依赖可分性的情况下，两种指数概念的一致性。

Maurey's factorization theory for operator spaces

We provide an operator space version of Maurey's factorization theorem. The main tool is an embedding result of independent interest. Applications for operator spaces and noncommutative Lp spaces are included.

A factorization constant for $l^n_p

We prove that if PT is a factorization of the identity operator on \ell_p^n through \ell_{\infty}^k, then ||P|| ||T|| \geq Cn^{1/p-1/2}(log n)^{-1/2}. This is a corollary of a more general result on factoring the identity operator on a quasi-normed space through \ell_{\infty}^k.

Factorization of the identity through operators with large diagonal

Given a Banach space~$X$ with an unconditional basis, we consider the following question: does the identity on~$X$ factor through every operator on~$X$ with large diagonal relative to the unconditional basis? We show that on Gowers' unconditional Banach space, there exists an operator for which the answer to the question is negative. By contrast, for any operator on the mixed-norm Hardy spaces $H^p(H^q)$, where $1 \leq p,q < \infty$, with the bi-parameter Haar system, this problem always has a positive solution. The spaces $L^p, 1 < p < \infty$, were treated first by Andrew~[{\em Studia Math.}…

Factorizations of EP operators

In this paper we characterize EP operators through the existence of different types of factorizations. Our results extend to EP operators existing characterizations for EP matrices and give new characterizations both for EP matrices and EP operators.

Factorizations and singular value estimates of operators with Gelfand-Shilov and Pilipović kernels

We prove that any linear operator with kernel in a Pilipović or Gelfand-Shilov space can be factorized by two operators in the same class. We also give links on numerical approximations for such compositions. We apply these composition rules to deduce estimates of singular values and establish Schatten-von Neumann properties for such operators.

Strongly irreducible factorization of quaternionic operators and Riesz decomposition theorem

Let $T$ be a bounded quaternionic normal operator on a right quaternionic Hilbert space $\mathcal{H}$. We show that $T$ can be factorized in a strongly irreducible sense, that is, for any $δ>0$ there exist a compact operator $K$ with $\|K\|< δ$, a partial isometry $W$ and a strongly irreducible operator $S$ on $\mathcal{H}$ such that \begin{equation*} T = (W+K) S. \end{equation*} We illustrate our result with an example. We also prove a quaternionic version of the Riesz decomposition theorem and as a consequence, show that if the spherical spectrum of a bounded quaternionic operator (need not …

On factorization of operators through the spaces $l^p.$

We give conditions on a pair of Banach spaces $X$ and $Y,$ under which each operator from $X$ to $Y,$ whose second adjoint factors compactly through the space $l^p,$ $1\le p\le+\infty$, itself compactly factors through $l^p.$ The conditions are as follows: either the space $X^*,$ or the space $Y^{***}$ possesses the Grothendieck approximation property. Leaving the corresponding question for parameters $p>1, p\neq 2,$ still open, we show that for $p=1$ the conditions are essential.

Toeplitz算子的分解

Factorization of Toeplitz operators

本文研究希尔伯特空间上交换压缩算子组诱导的Toeplitz算子，证明正T-Toeplitz算子及正纯下T-Toeplitz算子均可通过等距伪扩张进行分解，并进一步推广到(S,T)-Toeplitz算子情形。

Hilbert空间值高斯过程，其核、分解与协方差结构

Hilbert space valued Gaussian processes, their kernels, factorizations, and covariance structure

受应用启发，本文提出了算子值正定核的通用新框架，并将其应用于算子理论的膨胀构造以及构建Hilbert空间值高斯过程并分析其协方差结构。

球 Banach 函数空间上关于多重线性 Calderón-Zygmund 算子的 Hardy 空间弱分解

Weak Factorizations of the Hardy Spaces in Terms of Multilinear Calderón-Zygmund Operators on Ball Banach Function Spaces

本文在球 Banach 函数空间上基于多重线性 Calderón-Zygmund 算子建立了经典 Hardy 空间的弱分解，并通过该算子生成交换子的有界性获得了 BMO 空间的新刻画，结果可应用于多种函数空间。

Factorization of Asplund operators

We give necessary and sufficient conditions for an operator $A:X\to Y$ on a Banach space having a shrinking FDD to factor through a Banach space $Z$ such that the Szlenk index of $Z$ is equal to the Szlenk index of $A$. We also prove that for every ordinal $ξ\in (0, ω_1)\setminus\{ω^η: η<ω_1\text{\ a limit ordinal}\}$, there exists a Banach space $\mathfrak{G}_ξ$ having a shrinking basis and Szlenk index $ω^ξ$ such that for any separable Banach space $X$ and any operator $A:X\to Y$ having Szlenk index less than $ω^ξ$, $A$ factors through a subspace and through a quotient of $\mathfrak{G}_ξ$, a…

Factorization of operators on $C^*$-algebras

Let $A$ be a $C^*$-algebra. It is shown that every absolutely summing operator from $A$ into $\ell_2$ factors through a Hilbert space operator that belongs to the 4-Schatten- von Neumann class. We also provide finite dimensinal examples that show that one can not improve the 4-Schatten-von Neumann class to $p$-Schatten von Neumann class for any $p<4$. As application, we prove that there exists a modulus of capacity $ε\to N(ε)$ so that if $A$ is a $C^*$-algebra and $T \in Π_1(A,\ell_2)$ with $π_1(T)\leq 1$, then for every $ε>0$, the $ε$-capacity of the image of the unit ball of $A$ under $T$ do…

Wiener-Hopf operators admit triangular factorization

We prove that every positive bounded invertible Wiener-Hopf operator admits triangular factorization. This answers the question posed by L.Sakhnovich in 1994.

Separable morphisms of operator Hilbert systems, Pietsch factorizations and entanglement breaking maps

In this paper we investigate operator Hilbert systems and their separable morphisms. We prove that the operator Hilbert space of Pisier is an operator system, which possesses the self-duality property. It is established a link between unital positive maps and Pietch factorizations, which allows us to describe all separable morphisms from an abelian C*-algebra to an operator Hilbert system. Finally, we prove a key property of entanglement breaking maps that involves operator Hilbert systems.

Weak factorizations of the Hardy space in terms of multilinear fractional integral operator

We give a constructive proof of the factorization theorem for the classical Hardy space in terms of fractional integral operator. Moreover, the result is extended to the multilinear case and weighted case. As an application, we obtain the characterization of $BMO$ via the weighted boundedness of commutators of the multilinear fractional integral operator, without individual conditions on the weights class.

The factorizations of $H^ρ(\mathbb{R}^n)$ via multilinear Calderón-Zygmund operators on weighted Lebesgue spaces

We extend the recently much-studied Hardy factorization theorems to the weight case. The key point of this paper is to establish the factorization theorems without individual condition on the weight functions. As a direct application, we obtain the characterizations of $BMO(\mathbb{R}^n)$ space and Lipschitz spaces via the weighted boundedness of commutators of multilinear Calderón-Zygmund operators with the genuinely multilinear weights.

Factorization of Dirac operators on almost-regular fibrations of spin$^c$ manifolds

We establish the factorization of the Dirac operator on an almost-regular fibration of spin$^c$ manifolds in unbounded KK-theory. As a first intermediate result we establish that any vertically elliptic and symmetric first-order differential operator on a proper submersion defines an unbounded Kasparov module, and thus represents a class in KK-theory. Then, we generalize our previous results on factorizations of Dirac operators to proper Riemannian submersions of spin$^c$ manifolds. This allows us to show that the Dirac operator on the total space of an almost-regular fibration can be written …

Dimension dependence of factorization problems: Hardy spaces and $SL_n^\infty$

Given $1 \leq p < \infty$, let $W_n$ denote the finite-dimensional dyadic Hardy space $H_n^p$, its dual or $SL_n^\infty$. We prove the following quantitative result: The identity operator on $W_n$ factors through any operator $T : W_N\to W_N$ which has large diagonal with respect to the Haar system, where $N$ depends \emph{linearly} on $n$.

Completion, extension, factorization, and lifting of operators with a negative index

The famous results of M.G. Kre\uın concerning the description of selfadjoint contractive extensions of a Hermitian contraction $T_1$ and the characterization of all nonnegative selfadjoint extensions $\widetilde A$ of a nonnegative operator $A$ via the inequalities $A_K\leq \widetilde A \leq A_F$, where $A_K$ and $A_F$ are the Kre\uın-von Neumann extension and the Friedrichs extension of $A$, are generalized to the situation, where $\widetilde A$ is allowed to have a fixed number of negative eigenvalues. These generalizations are shown to be possible under a certain minimality condition on the…

Douglas' factorization theorem and atomic system in Hilbert pro$-C^{\ast}-$module

In the present paper we introduce the generalized inverse operators which have an interesting role in operator theory. We establish Douglas' factorization theorem type for Hilbert pro-$C^{\ast}$-module. We introduce the notion of atomic system and of $K$-frame in Hilbert pro-$C^{\ast}$-module and we study the relationship between them. We also demonstrat some properties of $K$-frame by using Douglas' factorization theorem.Finally we demonstrate that the sum of two $K$-frames in a Hilbert pro-$C^{\ast}$-module with certain conditions is once again a $K$-frame.

Effective construction of a class of positive operators in Hilbert space, which do not admit triangular factorization

A class of non-factorable positive operators is constructed. As a result, pure existence theorems in the well-known problems by Ringrose, Kadison and Singer are substituted by concrete examples.

驯服分解性质 I

Tame Factorization Property I

该研究为弗雷歇空间三元组引入驯服分解性质，通过算子半范数估计完成刻画，针对克塞空间三元组给出显式条件，还证明其与拟对角算子乘积构成的算子族的驯服性等价。

Kernels of unbounded Toeplitz operators and factorization of symbols

We consider kernels of unbounded Toeplitz operators in $H^p(\mathbb C^+)$ in terms of a factorization of their symbols. We study the existence of a minimal Toeplitz kernel containing a given function in $H^p(\mathbb C^+)$, we describe the kernels of Toeplitz operators whose symbol possesses a certain factorization involving two different Hardy spaces and we establish relations between the kernels of two operators whose symbols differ by a factor which corresponds, in the unit circle, to a non-integer power of $z$. We apply the results to describe the kernels of Toeplitz operators with non-vani…

A factorization of a quadratic pencils of accretive operators and applications

A canonical factorization is given for a quadratic pencil of accretive operators in a Hilbert space. Also, we establish some relationships between an m-accretive operator and its Moore-Penorse inverse. As an application, we study a result of existence, uniqueness, and maximal regularity of the strict solution for complete abstract second order differential equation in the non-homogeneous case. The paper is concluded with some questions left open from the preceding discussions.

Factorization Theorem for the Transfer Function Associated with an Unbounded Non-Self-Adjoint 2x2 Operator Matrix

We construct operators which factorize the transfer function associated with a non-self-adjoint 2x2 operator matrix whose diagonal entries can have overlapping spectra and whose off-diagonal entries are unbounded operators.

Factorization, majorization, and domination for linear relations

Let ${\mathfrak H}_A$, ${\mathfrak H}_B$, and ${\mathfrak H}$ be Hilbert spaces. Let $A$ be a linear relation from ${\mathfrak H}$ to ${\mathfrak H}_A$ and let $B$ be a linear relation from ${\mathfrak H}$ to ${\mathfrak H}_B$. If there exists an operator $Z \in \mathbf{B}({\mathfrak H}_B,{\mathfrak H}_A)$ such that $ZB \subset A$, then $B$ is said to dominate $A$. This notion plays a major role in the theory of Lebesgue type decompositions of linear relations and operators. There is a strong connection to the majorization and factorization in the well-known lemma of Douglas, when put in the c…

Effective Construction of a Positive Operator which does not admit Triangular Factorization

We have constructed a concrete example of a non-factorable positive operator. As a result, for the well-known problems (Ringrose, Kadison and Singer problems) we replace existence theorems by concrete examples.

Isometric factorization of vector measures and applications to spaces of integrable functions

Let $X$ be a Banach space, $Σ$ be a $σ$-algebra, and $m:Σ\to X$ be a (countably additive) vector measure. It is a well known consequence of the Davis-Figiel-Johnson-Pelczýnski factorization procedure that there exist a reflexive Banach space $Y$, a vector measure $\tilde{m}:Σ\to Y$ and an injective operator $J:Y \to X$ such that $m$ factors as $m=J\circ \tilde{m}$. We elaborate some theory of factoring vector measures and their integration operators with the help of the isometric version of the Davis-Figiel-Johnson-Pelczýnski factorization procedure. Along this way, we sharpen a result of Okad…

Strong factorizations of operators with applications to Fourier and Cesáro transforms

Consider two continuous linear operators $T\colon X_1(μ)\to Y_1(ν)$ and $S\colon X_2(μ)\to Y_2(ν)$ between Banach function spaces related to different $σ$-finite measures $μ$ and $ν$. We characterize by means of weighted norm inequalities when $T$ can be strongly factored through $S$, that is, when there exist functions $g$ and $h$ such that $T(f)=gS(hf)$ for all $f\in X_1(μ)$. For the case of spaces with Schauder basis our characterization can be improved, as we show when $S$ is for instance the Fourier operator, or the Cesàro operator. Our aim is to study the case when the map $T$ is besides…

与测试函数相关的 Schur-Agler 类中函数的分解

Factorization of functions in the Schur-Agler class related to test functions

本文研究与测试函数族相关的算子值函数何时可分解，并在多种情形下给出充要条件。

算子系统的近似因子分解性质

Approximate factorization properties for operator systems

本文证明算子系统张量积层次中的多种核性性质可表示为近似因子分解性质，推广了Choi-Effros的C*-代数完全正逼近性质及Han-Paulsen的算子系统核性性质。

Wiener空间变换的时间序指数分解

Factorization of Time-Ordered Exponentials for Wiener Space Transformations

提出算子代数框架，通过湮灭和产生算子将Wiener空间变换表示为时间序指数，并分解为行列式、乘法算子和平移算子，恢复经典Ramer-Kusuoka公式。

Approximation by analytic operator functions. Factorizations and very badly approximable functions

This is a continuation of our earlier paper \cite{PT3}. We consider here operator-valued functions (or infinite matrix functions) on the unit circle $\T$ and study the problem of approximation by bounded analytic operator functions. We discuss thematic and canonical factorizations of operator functions and study badly approximable and very badly approximable operator functions. We obtain algebraic and geometric characterizations of badly approximable and very badly approximable operator functions. Note that there is an important difference between the case of finite matrix functions and the ca…

Factorizations and invariant subspaces for weighted Schur classes

We study factorizations of operator valued functions of weighted Schur classes over multiply-connected domains. There is a correspondence between functions from weighted Schur classes and so-called ``conservative curved'' systems introduced in the paper. We show that the fundamental relationship between invariant subspaces of the main operator of a conservative system and factorizations of the corresponding operator valued function of Schur class, which is well known in the case of the unit disk, can be extended to our case. We develop new notions and constructions and discuss changes that sho…

Factorization of completely bounded maps through reflexive operator spaces with applications to weak almost periodicity

Let $(M,Γ)$ be a Hopf--von Neumann algebra, so that $M_\ast$ is a completely contractive Banach algebra. We investigate whether the product of two elements of $M$ that are both weakly almost periodic functionals on $M_\ast$ is again weakly almost periodic. For that purpose, we establish the following factorization result: If $M$ and $N$ are injective von Neumann algebras, and if $x, y \in M \bar{\otimes} N$ correspond to weakly compact operators from $M_\ast$ to $N$ factoring through reflexive operator spaces $X$ and $Y$, respectively, then the operator corresponding to $xy$ factors through th…

混合 Douglas 与弱优超分解定理

Mixing Douglas' and weak majorization and factorization theorems

本文提出了 Douglas 优超与分解定理及算子核包含关系的“混合”版本，适用于余域在积空间中的算子，并展示了其在线性偏微分方程耦合系统控制理论中的应用。

An extension of Nikishin's factorization theorem

A Nikishin-Maurey characterization is given for bounded subsets of weak-type Lebesgue spaces. New factorizations for linear and multilinear operators are shown to follow.

Factorization of a class of Toeplitz + hankel operators and the A_p-condition

Let $M(ϕ)=T(ϕ)+H(ϕ)$ be the Toeplitz plus Hankel operator acting on $H^p(\T)$ with generating function $ϕ\in L^\iy(\T)$. In a previous paper we proved that $M(ϕ)$ is invertible if and only if $ϕ$ admits a factorization $ϕ(t)=ϕ_{-}(t)ϕ_{0}(t)$ such that $ϕ_{-}$ and $ϕ_{0}$ and their inverses belong to certain function spaces and such that a further condition formulated in terms of $ϕ_{-}$ and $ϕ_{0}$ is satisfied. In this paper we prove that this additional condition is equivalent to the Hunt-Muckenhoupt-Wheeden condition (or, $A_{p}$-condition) for a certain function $σ$ defined on $[-1,1]$, w…

On Factorizations of Analytic Operator-Valued Functions and Eigenvalue Multiplicity Questions

We study several natural multiplicity questions that arise in the context of the Birman-Schwinger principle applied to non-self-adjoint operators. In particular, we re-prove (and extend) a recent result by Latushkin and Sukhtyaev by employing a different technique based on factorizations of analytic operator-valued functions due to Howland. Factorizations of analytic operator-valued functions are of particular interest in themselves and again we re-derive Howland's results and subsequently extend them. Considering algebraic multiplicities of finitely meromorphic operator-valued functions, we r…

与双线性形式和 Schur 乘子相关的唯一矩阵分解

Unique Matrix Factorizations associated to Bilinear Forms and Schur Multipliers

本文利用算子空间理论研究复 m×n 矩阵的 Grothendieck 型范数最优分解，证明四类分解中三类唯一、一类在特定条件下唯一。