Invertibility of Operator Matrices on a Banach Space

Let $\mathcal X $ and $\mathcal Y $ be Banach spaces, and let $A\in \mathcal B (\mathcal X )$ and $C\in \mathcal B (\mathcal Y , \mathcal X )$ be given operators. A necessary and sufficient condition is given for $\left[ \begin{array}{cc} A&C \\ X&Y \\ \end{array} \right]$ to be invertible (respectively, left invertible) for some $X\in \mathcal B (\mathcal X , \mathcal Y )$ and $Y\in \mathcal B (\mathcal Y )$ . Furthermore, some related results are obtained.     

Weighted Bloch spaces which are Banach spaces

Let $\mathcal{B }_\omega $ be a weighted Bloch space on the open unit disc which is a Banach space. In this paper, we study $\mathcal{B }_\omega $ by using four operators, that is, a point derivation, a point evaluation, a composition operator and an integral operator.     

Closed Range Property for Holomorphic Semi-Fredholm Functions

Given Banach spaces X and Y, we show that, for each operator-valued analytic map ${\alpha \in \mathcal O (D,\mathcal L(Y,X))}$ satisfying the finiteness condition ${\dim (X/\alpha (z)Y) < \infty}$ pointwise on an open set D in ${\mathbb {C}^n}$ , the induced multiplication operator ${\mathcal O(U,Y) \stackrel{\alpha}{\longrightarrow} \mathcal O (U,X)}$ has closed range on each Stein open set ${U \subset D}$ . As an application we deduce that the generalized range ${{\rm R}^{\infty}(T) = \bigcap_{k \geq 1}\sum_{| \alpha | = k} T^{\alpha}X}$ of a commuting multioperator ${T \in \mathcal L(X)^n}$ with ${\dim(X/\sum_{i=1}^n T_iX) < \infty}$ can be represented as a suitable spectral subspace.     

A unified approach for minimizing composite norms

We propose a first-order augmented Lagrangian algorithm (FALC) to solve the composite norm minimization problem $$\begin{aligned} \begin{array}{ll} \min \limits _{X\in \mathbb{R }^{m\times n}}&\mu _1\Vert \sigma (\mathcal{F }(X)-G)\Vert _\alpha +\mu _2\Vert \mathcal{C }(X)-d\Vert _\beta ,\\ \text{ subject} \text{ to}&\mathcal{A }(X)-b\in \mathcal{Q }, \end{array} \end{aligned}$$ where $\sigma (X)$ denotes the vector of singular values of $X \in \mathbb{R }^{m\times n}$ , the matrix norm $\Vert \sigma (X)\Vert _{\alpha }$ denotes either the Frobenius, the nuclear, or the $\ell _2$ -operator norm of $X$ , the vector norm $\Vert .\Vert _{\beta }$ denotes either the $\ell _1$ -norm, $\ell _2$ -norm or the $\ell _{\infty }$ -norm; $\mathcal{Q }$ is a closed convex set and $\mathcal{A }(.)$ , $\mathcal{C }(.)$ , $\mathcal{F }(.)$ are linear operators from $\mathbb{R }^{m\times n}$ to vector spaces of appropriate dimensions. Basis pursuit, matrix completion, robust principal component pursuit (PCP), and stable PCP problems are all special cases of the composite norm minimization problem. Thus, FALC is able to solve all these problems in a unified manner. We show that any limit point of FALC iterate sequence is an optimal solution of the composite norm minimization problem. We also show that for all $\epsilon >0$ , the FALC iterates are $\epsilon $ -feasible and $\epsilon $ -optimal after $\mathcal{O }(\log (\epsilon ^{-1}))$ iterations, which require $\mathcal{O }(\epsilon ^{-1})$ constrained shrinkage operations and Euclidean projection onto the set $\mathcal{Q }$ . Surprisingly, on the problem sets we tested, FALC required only $\mathcal{O }(\log (\epsilon ^{-1}))$ constrained shrinkage, instead of the $\mathcal{O }(\epsilon ^{-1})$ worst case bound, to compute an $\epsilon $ -feasible and $\epsilon $ -optimal solution. To best of our knowledge, FALC is the first algorithm with a known complexity bound that solves the stable PCP problem.     

A Topologist’s View of Chu Spaces

For a symmetric monoidal-closed category $\mathcal{X}$ and any object K, the category of K-Chu spaces is small-topological over $\mathcal{X}$ and small cotopological over $\mathcal{X}^{{{\text{op}}}}$ . Its full subcategory of $\mathcal{M}$ -extensive K-Chu spaces is topological over $\mathcal{X}$ when $\mathcal{X}$ is $\mathcal{M}$ -complete, for any morphism class $\mathcal{M}$ . Often this subcategory may be presented as a full coreflective subcategory of Diers’ category of affine K-spaces. Hence, in addition to their roots in the theory of pairs of topological vector spaces (Barr) and their connections with linear logic (Seely), the Dialectica categories (Hyland, de Paiva), and with the study of event structures for modeling concurrent processes (Pratt), Chu spaces seem to have a less explored link with algebraic geometry. We use the Zariski closure operator to describe the objects of the *-autonomous category of $\mathcal{M}$ -extensive and $\mathcal{M}$ -coextensive K-Chu spaces in terms of Zariski separation and to identify its important subcategory of complete objects.     

On the unique representation of very strong algebraic geometry codes

This paper addresses the question of retrieving the triple ${(\mathcal X,\mathcal P, E)}$ from the algebraic geometry code ${\mathcal C = \mathcal C_L(\mathcal X, \mathcal P, E)}$ , where ${\mathcal X}$ is an algebraic curve over the finite field ${\mathbb F_q, \,\mathcal P}$ is an n-tuple of ${\mathbb F_q}$ -rational points on ${\mathcal X}$ and E is a divisor on ${\mathcal X}$ . If ${\deg(E)\geq 2g+1}$ where g is the genus of ${\mathcal X}$ , then there is an embedding of ${\mathcal X}$ onto ${\mathcal Y}$ in the projective space of the linear series of the divisor E. Moreover, if ${\deg(E)\geq 2g+2}$ , then ${I(\mathcal Y)}$ , the vanishing ideal of ${\mathcal Y}$ , is generated by ${I_2(\mathcal Y)}$ , the homogeneous elements of degree two in ${I(\mathcal Y)}$ . If ${n >2 \deg(E)}$ , then ${I_2(\mathcal Y)=I_2(\mathcal Q)}$ , where ${\mathcal Q}$ is the image of ${\mathcal P}$ under the map from ${\mathcal X}$ to ${\mathcal Y}$ . These three results imply that, if ${2g+2\leq m < \frac{1}{2}n}$ , an AG representation ${(\mathcal Y, \mathcal Q, F)}$ of the code ${\mathcal C}$ can be obtained just using a generator matrix of ${\mathcal C}$ where ${\mathcal Y}$ is a normal curve in ${\mathbb{P}^{m-g}}$ which is the intersection of quadrics. This fact gives us some clues for breaking McEliece cryptosystem based on AG codes provided that we have an efficient procedure for computing and decoding the representation obtained.     

Large free sets in powers of universal algebras

We prove that for each universal algebra ${(A, \mathcal{A})}$ of cardinality ${|A| \geq 2}$ and infinite set X of cardinality ${|X| \geq | \mathcal{A}|}$ , the X-th power ${(A^{X}, \mathcal{A}^{X})}$ of the algebra ${(A, \mathcal{A})}$ contains a free subset ${\mathcal{F} \subset A^{X}}$ of cardinality ${|\mathcal{F}| = 2^{|X|}}$ . This generalizes the classical Fichtenholtz–Kantorovitch–Hausdorff result on the existence of an independent family ${\mathcal{I} \subset \mathcal{P}(X)}$ of cardinality ${|\mathcal{I}| = |\mathcal{P}(X)|}$ in the Boolean algebra ${\mathcal{P}(X)}$ of subsets of an infinite set X.     

L 2-theory for non-symmetric Ornstein–Uhlenbeck semigroups on domains

We prove that the mild solution of the stochastic evolution equation ${{d}X(t) = AX(t)\,{d}t + {d}W(t)}$ on a Banach space E has a continuous modification if the associated Ornstein–Uhlenbeck semigroup is analytic on L ² with respect to the invariant measure. This result is used to extend recent work of Da Prato and Lunardi for Ornstein–Uhlenbeck semigroups on domains ${\mathcal{O} \subseteq E}$ to the non-symmetric case. Denoting the generator of the Ornstein–Uhlenbeck semigroup by ${L_\mathcal{O}}$ , we obtain sufficient conditions in order that the domain of ${\sqrt{-L_\mathcal{O}}}$ be a first-order Sobolev space.     

Conditional Reducibility of Certain Unbounded Nonnegative Hamiltonian Operator Functions

Let J and ${{\mathfrak{J}}}$ be operators on a Hilbert space ${{\mathcal{H}}}$ which are both self-adjoint and unitary and satisfy ${J{\mathfrak{J}}=-{\mathfrak{J}}J}$ . We consider an operator function ${{\mathfrak{A}}}$ on [0, 1] of the form ${{\mathfrak{A}}(t)={\mathfrak{S}}+{\mathfrak{B}}(t)}$ , ${t \in [0, 1]}$ , where ${\mathfrak{S}}$ is a closed densely defined Hamiltonian ( ${={\mathfrak{J}}}$ -skew-self-adjoint) operator on ${{\mathcal{H}}}$ with ${i {\mathbb{R}} \subset \rho ({\mathfrak{S}})}$ and ${{\mathfrak{B}}}$ is a function on [0, 1] whose values are bounded operators on ${{\mathcal{H}}}$ and which is continuous in the uniform operator topology. We assume that for each ${t \in [0,1] \,{\mathfrak{A}}(t)}$ is a closed densely defined nonnegative (=J-accretive) Hamiltonian operator with ${i {\mathbb{R}} \subset \rho({\mathfrak{A}}(t))}$ . In this paper we give sufficient conditions on ${{\mathfrak{S}}}$ under which ${{\mathfrak{A}}}$ is conditionally reducible, which means that, with respect to a natural decomposition of ${{\mathcal{H}}}$ , ${{\mathfrak{A}}}$ is diagonalizable in a 2×2 block operator matrix function such that the spectra of the two operator functions on the diagonal are contained in the right and left open half planes of the complex plane. The sufficient conditions involve bounds on the resolvent of ${{\mathfrak{S}}}$ and interpolation of Hilbert spaces.     

Ranks of backward shift invariant subspaces of the Hardy space over the bidisk

Let $\{\varphi _n(z)\}_{n\ge 0}$ be a sequence of inner functions satisfying that $\zeta _n(z):=\varphi _n(z)/\varphi _{n+1}(z)\in H^\infty (z)$ for every $n\ge 0$ and $\{\varphi _n(z)\}_{n\ge 0}$ has no nonconstant common inner divisors. Associated with it, we have a Rudin type invariant subspace $\mathcal{M }$ of $H^2(\mathbb{D }^2)$ . The ranks of $\mathcal{M }\ominus w\mathcal{M }$ for $\mathcal{F }_z$ and $\mathcal{F }^*_z$ respectively are determined, where $\mathcal{F }_z$ is the fringe operator on $\mathcal{M }\ominus w\mathcal{M }$ . Let $\mathcal{N }= H^2(\mathbb{D }^2)\ominus \mathcal{M }$ . It is also proved that the rank of $\mathcal{M }\ominus w\mathcal{M }$ for $\mathcal{F }^*_z$ equals to the rank of $\mathcal{N }$ for $T^*_z$ and $T^*_w$ .     

On the Generality of Assuming that a Family of Continuous Functions Separates Points

For an algebra ${\mathcal{A}}$ of complex-valued, continuous functions on a compact Hausdorff space (X, τ), it is standard practice to assume that ${\mathcal{A}}$ separates points in the sense that for each distinct pair ${x, y \in X}$ , there exists an ${f \in \mathcal{A}}$ such that ${f(x) \neq f(y)}$ . If ${\mathcal{A}}$ does not separate points, it is known that there exists an algebra ${\widehat{\mathcal{A}}}$ on a compact Hausdorff space ${(\widehat{X}, \widehat{\tau})}$ that does separate points such that the map ${\mathcal{A} \mapsto \widehat{\mathcal{A}}}$ is a uniform norm isometric algebra isomorphism. So it is, to a degree, without loss of generality that we assume ${\mathcal{A}}$ separates points. The construction of ${{\widehat{\mathcal{A}}}}$ and ${(\widehat{X}, \widehat{\tau})}$ does not require that ${\mathcal{A}}$ has any algebraic structure nor that ${(X, \tau)}$ has any properties, other than being a topological space. In this work we develop a framework for determining the degree to which separation of points may be assumed without loss of generality for any family ${\mathcal{A}}$ of bounded, complex-valued, continuous functions on any topological space ${(X, \tau)}$ . We also demonstrate that further structures may be preserved by the mapping ${\mathcal{A} \mapsto \widehat{\mathcal{A}}}$ , such as boundaries of weak peak points, the Lipschitz constant when the functions are Lipschitz on a compact metric space, and the involutive structure of real function algebras on compact Hausdorff spaces.     

Proper/Residually-Finite Idempotent Semirings

An idempotent semiring (= ISR) is called L-E if its underlying additive Abelian semigroup is generated by join-primes. Not all ISRs are L-E; not even when finite. The submodule of “linear-recognizable” elements of an ISR $\mathcal {M}$ is denoted $\mathcal {C}\mathcal {M}$ , and $\mathcal {M}$ is called proper if there are enough elements in $\mathcal {C}\mathcal {M}$ to separate points. If there are enough finite-index congruences to separate points, $\mathcal {M}$ is called residually-finite. Finite and proper ISRs are always residually-finite, but finite ISRs are not always proper, unless they are L-E. For certain classes of ISRs, conditions are given to guarantee proper and residual-finiteness. Among these is one which requires that the compact elements of the linear dual of $\mathcal {M}$ belong to $\mathcal {C}\mathcal {M}$ . Another condition requires that the recognizable subsets of a certain underlying monoid remain recognizable under the closure operator relative to a certain natural topology. These conditions are automatic for any finite L-E ISR, or any L-E ISR arising from a bounded, distributive lattice. Thus, a large class of proper/residually-finite ISRs exists. Moreover, the theorem of Malcev for semigroups (finitely-generated, commutative implies residually-finite) is shown to fail for ISRs in general.     

Existence, automatic continuity and invariant submodules of generalized derivations on modules

Let ${\mathcal{A}}$ be a ${\mathbb{C}}$ -algebra, δ be a derivation on ${\mathcal{A}}$ and ${\mathcal{M}}$ be a left ${\mathcal{A}}$ -module. A linear map ${\tau : \mathcal{M} \rightarrow \mathcal{M}}$ is called a generalized derivation relative to δ if ${\tau(am)=a\tau(m)+\delta(a)m\,(a \in \mathcal{A}, m \in \mathcal{M})}$ . In this article first we study the existence of generalized derivations. In particular we show that free modules and projective modules always have nontrivial generalized derivations relative to nonzero derivations of ${\mathcal{A}}$ . Then we investigate the invariance of prime submodules under generalized derivations. Specifically we show that every minimal prime submodule of ${\mathcal{M}}$ is invariant under every generalized derivation. Moreover we obtain analogs of Posner’s theorem for generalized derivations. In the case that ${\mathcal{A}}$ is a Banach algebra and ${\mathcal{M}}$ is a Banach left ${\mathcal{A}}$ -module, we study the existence of continuous generalized derivations and automatic continuity of generalized derivations.     

Generalized Complex Spherical Harmonics, Frame Functions, and Gleason Theorem

Consider a finite dimensional complex Hilbert space ${\mathcal{H}}$ , with ${dim(\mathcal{H}) \geq 3}$ , define ${\mathbb{S}(\mathcal{H}):= \{x\in \mathcal{H} \:|\: \|x\|=1\}}$ , and let ${\nu_\mathcal{H}}$ be the unique regular Borel positive measure invariant under the action of the unitary operators in ${\mathcal{H}}$ , with ${\nu_\mathcal{H}(\mathbb{S}(\mathcal{H}))=1}$ . We prove that if a complex frame function ${f : \mathbb{S}(\mathcal{H})\to \mathbb{C}}$ satisfies ${f \in \mathbb{L}^2(\mathbb{S}(\mathcal{H}), \nu_\mathcal{H})}$ , then it verifies Gleason’s statement: there is a unique linear operator ${A: \mathcal{H} \to \mathcal{H}}$ such that ${f(u) = \langle u| A u\rangle}$ for every ${u \in \mathbb{S}(\mathcal{H}).\,A}$ is Hermitean when f is real. No boundedness requirement is thus assumed on f a priori.     

Hypercentralizing generalized skew derivations on left ideals in prime rings

Let $\mathcal{R }$ be a prime ring of characteristic different from $2, \mathcal{Q }_r$ the right Martindale quotient ring of $\mathcal{R }, \mathcal{C }$ the extended centroid of $\mathcal{R }, \mathcal{I }$ a nonzero left ideal of $\mathcal{R }, F$ a nonzero generalized skew derivation of $\mathcal{R }$ with associated automorphism $\alpha $ , and $n,k \ge 1$ be fixed integers. If $[F(r^n),r^n]_k=0$ for all $r \in \mathcal{I }$ , then there exists $\lambda \in \mathcal{C }$ such that $F(x)=\lambda x$ , for all $x\in \mathcal{I }$ . More precisely one of the following holds: (1) $\alpha $ is an $X$ -inner automorphism of $\mathcal{R }$ and there exist $b,c \in \mathcal{Q }_r$ and $q$ invertible element of $\mathcal{Q }_r$ , such that $F(x)=bx-qxq^{-1}c$ , for all $x\in \mathcal{Q }_r$ . Moreover there exists $\gamma \in \mathcal{C }$ such that $\mathcal{I }(q^{-1}c-\gamma )=(0)$ and $b-\gamma q \in \mathcal{C }$ ; (2) $\alpha $ is an $X$ -outer automorphism of $\mathcal{R }$ and there exist $c \in \mathcal{Q }_r, \lambda \in \mathcal{C }$ , such that $F(x)=\lambda x-\alpha (x)c$ , for all $x\in \mathcal{Q }_r$ , with $\alpha (\mathcal{I })c=0$ .     

Algebras of Convolution Type Operators with Piecewise Slowly Oscillating Data. II: Local Spectra and Fredholmness

Let ${\mathcal{B}_{p,w}}$ be the Banach algebra of all bounded linear operators acting on the weighted Lebesgue space ${L^p(\mathbb{R},w)}$ , where ${p\in(1,\infty)}$ and w is a Muckenhoupt weight. We study the Banach subalgebra ${\mathfrak{U}_{p,w}}$ of ${\mathcal{B}_{p,w}}$ generated by all multiplication operators aI ( ${a\in PSO^\diamond}$ ) and all convolution operators W ⁰(b) ( ${b\in PSO_{p,w}^\diamond}$ ), where ${PSO^\diamond\subset L^\infty(\mathbb{R})}$ and ${PSO_{p,w}^\diamond\subset M_{p,w}}$ are algebras of piecewise slowly oscillating functions that admit piecewise slowly oscillating discontinuities at arbitrary points of ${\mathbb{R}\cup\{\infty\}}$ , and M _p,w is the Banach algebra of Fourier multipliers on ${L^p(\mathbb{R},w)}$ . Under some conditions on the Muckenhoupt weight w, using results of the local study of ${\mathfrak{U}_{p,w}}$ obtained in the first part of the paper and applying the theory of Mellin pseudodifferential operators and the two idempotents theorem, we now construct a Fredholm symbol calculus for the Banach algebra ${\mathfrak{U}_{p,w}}$ and establish a Fredholm criterion for the operators ${A\in\mathfrak{U}_{p,w}}$ in terms of their Fredholm symbols. In four partial cases we obtain for ${\mathfrak{U}_{p,w}}$ more effective results.     

A Characterization of Compact Operators Via the Non-Connectedness of the Attractors of a Family of IFSs

In this paper we present a result which establishes a connection between the theory of compact operators and the theory of iterated function systems. For a Banach space $X$ , $S$ and $T$ bounded linear operators from $X$ to $X$ such that $\Vert S\Vert , \Vert T\Vert <1$ and $w\in X$ , let us consider the IFS $\mathcal S _{w}=(X,f_{1},f_{2})$ , where $f_{1},f_{2}:X\rightarrow X$ are given by $f_{1}(x)=S(x)$ and $f_{2}(x)=T(x)+w$ , for all $x\in X$ . On one hand we prove that if the operator $S$ is compact, then there exists a family $(K_{n})_{n\in \mathbb N }$ of compact subsets of $X$ such that $A_{\mathcal S _{w}}$ is not connected, for all $w\in X-\bigcup _{n\in \mathbb N } K_{n}$ . On the other hand we prove that if $H$ is an infinite dimensional Hilbert space, then a bounded linear operator $S:H\rightarrow H$ having the property that $\Vert S\Vert <1$ is compact provided that for every bounded linear operator $T:H\rightarrow H$ such that $\Vert T\Vert <1$ there exists a sequence $(K_{T,n})_{n}$ of compact subsets of $H$ such that $A_{\mathcal S _{w}}$ is not connected for all $w\in H-\bigcup _{n}K_{T,n}$ . Consequently, given an infinite dimensional Hilbert space $H$ , there exists a complete characterization of the compactness of an operator $S:H\rightarrow H$ by means of the non-connectedness of the attractors of a family of IFSs related to the given operator. Finally we present three examples illustrating our results.     

Algebras of Convolution Type Operators with Piecewise Slowly Oscillating Data. I: Local and Structural Study

Let ${\mathcal{B}_{p,w}}$ be the Banach algebra of all bounded linear operators acting on the weighted Lebesgue space ${L^{p}(\mathbb{R}, w)}$ , where ${p \in (1, \infty)}$ and w is a Muckenhoupt weight. We study the Banach subalgebra ${\mathfrak{A}_{p,w}}$ of ${\mathcal{B}_{p,w}}$ generated by all multiplication operators aI ( ${a \in PSO^{\diamond}}$ ) and all convolution operators W ⁰(b) ( ${b \in PSO_{p,w}^{\diamond}}$ ), where ${PSO^{\diamond} \subset L^{\infty}(\mathbb{R})}$ and ${PSO_{p,w}^{\diamond} \subset M_{p,w}}$ are algebras of piecewise slowly oscillating functions that admit piecewise slowly oscillating discontinuities at arbitrary points of ${\mathbb{R} \cup \{\infty\}}$ , and M _p,w is the Banach algebra of Fourier multipliers on ${L^{p}(\mathbb{R}, w)}$ . Under some conditions on the Muckenhoupt weight w, we construct a Fredholm symbol calculus for the Banach algebra ${\mathfrak{A}_{p,w}}$ and establish a Fredholm criterion for the operators ${A \in \mathfrak{A}_{p,w}}$ in terms of their Fredholm symbols. To study the Banach algebra ${\mathfrak{A}_{p,w}}$ we apply the theory of Mellin pseudodifferential operators, the Allan–Douglas local principle, the two idempotents theorem and the method of limit operators. The paper is divided in two parts. The first part deals with the local study of ${\mathfrak{A}_{p,w}}$ and necessary tools for studying local algebras.     

The Calabi homomorphism, Lagrangian paths and special Lagrangians

Let $\mathcal{O }$ be an orbit of the group of Hamiltonian symplectomorphisms acting on the space of Lagrangian submanifolds of a symplectic manifold $(X,\omega ).$ We define a functional $\mathcal{C }:\mathcal{O } \rightarrow \mathbb{R }$ for each differential form $\beta $ of middle degree satisfying $\beta \wedge \omega = 0$ and an exactness condition. If the exactness condition does not hold, $\mathcal{C }$ is defined on the universal cover of $\mathcal{O }.$ A particular instance of $\mathcal{C }$ recovers the Calabi homomorphism. If $\beta $ is the imaginary part of a holomorphic volume form, the critical points of $\mathcal{C }$ are special Lagrangian submanifolds. We present evidence that $\mathcal{C }$ is related by mirror symmetry to a functional introduced by Donaldson to study Einstein–Hermitian metrics on holomorphic vector bundles. In particular, we show that $\mathcal{C }$ is convex on an open subspace $\mathcal{O }^+ \subset \mathcal{O }.$ As a prerequisite, we define a Riemannian metric on $\mathcal{O }^+$ and analyze its geodesics. Finally, we discuss a generalization of the flux homomorphism to the space of Lagrangian submanifolds, and a Lagrangian analog of the flux conjecture.