Let $M_{z}$ be the multiplication operator on the Bergman space and $M_{I}$ denote the restriction of $M_{z}$ to an invariant subspace $I$. A question raised by K. Zhu is that when are two restriction operators $M_{I}$ and $M_{J}$ are similar? In this note, we give some sufficient conditions of this problem in a general case.
Let $X$ be a Banach space, $A\in B(X)$ and $M$ be an invariant subspace of $A$. We present an alternative proof that, if the spectrum of the restriction of $A$ to $M$ contains a point that is in any given hole in the spectrum of $A$, then the entire hole is in the spectrum of the restriction.
In this paper we consider a problem of the similarity of complex symmetric operators to perturbations of restrictions of normal operators. For a subclass of cyclic complex symmetric operators in a finite-dimensional Hilbert space we prove the similarity to rank-one perturbations of restrictions of normal operators. The main tools are a truncated moment problem in $\mathbb{C}$, and some objects similar to objects from the theory of spectral problems for Jacobi matrices.
The Lifting Problem for Commuting Subnormals (LPCS) asks for necessary and sufficient conditions for a pair of subnormal operators on Hilbert space to admit commuting normal extensions. \ We study LPCS within the class of commuting 2-variable weighted shifts $\mathbf{T} \equiv (T_1,T_2)$ with subnormal components $T_1$ and $T_2$, acting on the Hilbert space $\ell ^2(\mathbb{Z}^2_+)$ with canonical orthonormal basis $\{e_{(k_1,k_2)}\}_{k_1,k_2 \geq 0}$ . \ The \textit{core} of a commuting 2-variable weighted shift $\mathbf{T}$, $c(\mathbf{T})$, is the restriction of $\mathbf{T}$ to the invarian…
We provide a direct, intersection theoretic, argument that the Jordan models of an operator of class C_{0}, of its restriction to an invariant subspace, and of its compression to the orthogonal complement, satisfy a multiplicative form of the Horn inequalities, where `inequality' is replaced by `divisibility'. When one of these inequalities is saturated, we show that there exists a splitting of the operator into quasidirect summands which induces similar splittings for the restriction of the operator to the given invariant subspace and its compression to the orthogonal complement. The result i…
It is well known that for a single bounded operator $A_0$ on a Hilbert $\mathfrak{H}$, if $\mathfrak{M}\subset \mathfrak{H}$ is hyperinvariant for $A_0$, then the spectrum of $A_0|_{\mathfrak{M}}$ is contained in the spectrum of $A_0$. In this note, we modify an example of Taylor to prove the following. There exist a quadruple $A=(A_1,A_2,A_3,A_4)$ of commuting bounded Hilbert space operators and a hyperinvariant subspace $\mathfrak{X}_1$ for $A$ such that the Taylor joint spectrum of $A$ restricted to $\mathfrak{X}_1$ is a not a subset of the Taylor joint spectrum of $A$.
We relax assumptions for a dissipative operator in Krein space to possess a maximal non-negative invariant subspace. Our main result is a generalization of a well-known Pontrjagin-Krein-Langer-Azizov theorem. Then we investigate the semigroup properties of the restriction of the operator onto the invariant subspace.
A frame is a system of vectors $S$ in Hilbert space $\mathscr{H}$ with properties which allow one to write algorithms for the two operations, analysis and synthesis, relative to $S$, for all vectors in $\mathscr{H}$; expressed in norm-convergent series. Traditionally, frame properties are expressed in terms of an $S$-Gramian, $G_{S}$ (an infinite matrix with entries equal to the inner product of pairs of vectors in $S$); but still with strong restrictions on the given system of vectors in $S$, in order to guarantee frame-bounds. In this paper we remove these restrictions on $G_{S}$, and we obt…
We describe necessary and sufficient conditions for a $J$-dissipative operator in a Krein space to have maximal semidefinite invariant subspaces. The semigroup properties of the restrictions of an operator to these subspaces are studied. Applications are given to the case when an operator admits matrix representation with respect to the canonical decomposition of the space. The main conditions are given in the terms of the interpolation theory of Banach spaces.
The theorem on the existence of maximal nonnegative invariant subspaces for a special class of dissipative operators in Hilbert space with indefinite inner product is proved in the paper. It is shown in addition that the spectra of the restrictions of these operators on the corresponding invariant subspaces lie in the closed upper half-plane. The obtained theorem is a generalization of well-known results of L.S.Pontrjagin, H.K.Langer, M.G.Krein and T.Ja.Azizov devoted to this subject.
Doubly commutativity of invariant subspaces of the Bergman space and the Dirichlet space over the unit polydisc $\mathbb{D}^n$ (with $ n \geq 2$) is investigated. We show that for any non-empty subset $α=\{α_1,\dots,α_k\}$ of $\{1,\dots,n\}$ and doubly commuting invariant subspace $\s$ of the Bergman space or the Dirichlet space over $\D^n$, the tuple consists of restrictions of co-ordinate multiplication operators $M_α|_\s:=(M_{z_{α_1}}|_\s,\dots, M_{z_{α_k}}|_\s)$ always possesses wandering subspace of the form \[\bigcap_{i=1}^k(\s\ominus z_{α_i}\s). \]
For an operator $T$ of class $C_0$ with multiplicity two, we show that the quasisimilarity class of an invariant subspace $M$ is determined by the quasisimilarity classes of the restriction $T|M$ and of the compression $T_{M^\perp}$. We also provide a canonical form for the subspace $M$.
We introduce and study multivariate generalizations of the classical BV spaces of Jordan, F. Riesz and Wiener. The family of the introduced spaces contains or is intimately related to a considerable class of function spaces of modern analysis including BMO, BV, Morrey spaces and those of Sobolev of arbitrary smoothness, Besov and Triebel-Lizorkin spaces. We prove under mild restrictions that the BV spaces of this family are dual and present constructive characterizations of their preduals via atomic decompositions. Moreover, we show that under additional restrictions such a predual space is is…
We prove that if T is a strictly singular 1-1 operator defined on an infinite dimensional Banach space X, then for every infinite dimensional subspace Y of X there exists an infinite dimensional subspace Z of Y such that Z contains orbits of T of every finite length and the restriction of T on Z is a compact operator.
In our companion paper (S.N. Chandler Wilde, D.P. Hewett, A. Moiola, Sobolev spaces on non-Lipschitz subsets of $\mathbb{R}^n$ with application to boundary integral equations on fractal screens, 2016) we studied a number of different Sobolev spaces on a general (non-Lipschitz) open subset $Ω$ of $\mathbb{R}^n$, defined as closed subspaces of the classical Bessel potential spaces $H^s(\mathbb{R}^n)$ for $s\in\mathbb{R}$. These spaces are mapped by the restriction operator to certain spaces of distributions on $Ω$. In this note we make some observations about the relation between these spaces of…
We consider the restriction of twice differentiable functionals on a Hilbert space to families of subspaces that vary continuously with respect to the gap metric. We study bifurcation of branches of critical points along these families, and apply our results to semilinear systems of ordinary differential equations.
An operator $T$ from a Banach lattice $E$ into a Banach space is disjointly non-singular ($DN$-$S$, for short) if no restriction of $T$ to a subspace generated by a disjoint sequence is strictly singular. We obtain several results for $DN$-$S$ operators, including a perturbative characterization. For $E=L_p$ ($1< p<\infty$) we improve the results, and we show that the $DN$-$S$ operators have a different behavior in the cases $p=2$ and $p\neq 2$. As an application we prove that the strongly embedded subspaces of $L_p$ form an open subset in the set of all closed subspaces.
It is shown that under certain stability conditions a complemented subspace of the space $s$ of rapidly decreasing sequences is isomorphic to $s$ and this condition characterizes $s$. This result is used to show that for the classical Cantor set $X$ the space $C_\infty(X)$ of restrictions to $X$ of $C^\infty$-functions on $\R$ is isomorphic to $s$, so completing the theory developed in "Restriction spaces of $A^\infty$", to appear in Rev. Mat. Iberoamericana 29.4 (2013)
We answer the question of W.T. Gowers, giving an example of a bounded operator on a subspace of Gowers unconditional space which is not a strictly singular perturbation of a restriction of a diagonal operator. We make some observations on operators in arbitrary tight by support Banach space, showing in particular that in such space no two isomorphic infinitely dimensional subspaces form a direct sum.
We consider the problem of classification of invariant subspaces for the class of uniform Jordan operators. We show that given two invariant subspaces $M_1$ and $M_2$ of a uniform Jordan operator $T=S(θ)\oplus S(θ)\oplus \ldots$, the subspace $M_2$ belongs to the quasiaffine orbit of $M_1$ if and only if the restrictions $T|M_1$ and $T|M_2$ are quasisimilar and the compression $T_{M_2^\perp}$ can be injected in the compression $T_{M_1^\perp}$. Our result refines previous work on the subject by Bercovici and Smotzer.
Let $T$ be a polynomially bounded operator, and let $\mathcal M$ be its invariant subspace. Suppose that $P_{\mathcal M^\perp}T|_{\mathcal M^\perp}$ is similar to a contraction, while $θ(T|_{\mathcal M})=0$, where $θ$ is a finite product of Blaschke products with simple zeros satisfying the Carleson interpolating condition (a Carleson--Newman Blaschke product). Then $T$ is similar to a contraction. It is mentioned that Le Merdy's example shows that the assumption of polynomially boundedness cannot be replaced by the assumption of power boundedness.
The Invariant Subspace Problem ("ISP") for Hilbert space operators is known to be equivalent to a question that, on its surface, seems surprisingly concrete: For composition operators induced on the Hardy space H^2 by hyperbolic automorphisms of the unit disc, is every nontrivial minimal invariant subspace one dimensional (i.e., spanned by an eigenvector)? In the hope of reviving interest in the contribution this remarkable result might offer to the studies of both composition operators and the ISP, I revisit some known results, weaken their hypotheses and simplify their proofs. Sample results…
This paper addresses the problem of improving properties of a linear operator u in $l_2^n$ by restricting it onto coordinate subspaces. We discuss how to reduce the norm of u by a random coordinate restriction, how to approximate u by a random operator with small "coordinate" rank, how to find coordinate subspaces where u is an isomorphism. The first problem in this list provides a probabilistic extension of a suppression theorem of Kashin and Tzafriri, the second one is a new look at a result of Rudelson on the random vectors in the isotropic position, the last one is the recent generalizatio…
The work is devoted to the study of the similarity of a correct restriction to some self-adjoint operator in the case when the minimal operator is symmetric. The resulting theorem was applied to the Sturm-Liouville operator and the Laplace operator. It is shown that the spectrum of a non self-adjoint singularly perturbed operator is real and the corresponding system of eigenvectors forms a Riesz basis.
Let $S$ be a subspace of $L^2 (\bm{R})$. We show that the operator $M$ of multiplication by the independent variable has a simple symmetric regular restriction to $S$ with deficiency indices $(1,1)$ if and only if $S = u h K^{2}_θ$ is a nearly invariant subspace, with $θ$ a meromorphic inner function vanishing at $i$. Here $u$ is unimodular, $h$ is an isometric multiplier of $K^{2}_θ$ into $H^2$ and $H^2$ is the Hardy space of the upper half plane. Our proof uses the dilation theory of completely positive maps.
An operator $T$ is called a 3-isometry if there exists operators $B_1(T^*,T)$ and $B_2(T^*,T)$ such that \[Q(n)=T^{*n}T^n=1+nB_1(T^*,T)+n^2 B_2(T^*,T)\] for all natural numbers $n$. An operator $J$ is a Jordan operator of order $2$ if $J=U+N$ where $U$ is unitary, $N$ is nilpotent order $2$, and $U$ and $N$ commute. An easy computation shows that $J$ is a $3$-isometry and that the restriction of $J$ to an invariant subspace is also a $3$-isometry. Those $3$-isometries which are the restriction of a Jordan operator to an invariant subspace can be identified, using the theory of completely posit…
We show that a bounded analytic semigroup on an $L_p$-space has a bounded $H^{\infty}(Σ_φ)$-calculus for some $φ< \fracπ{2}$ if and only if the semigroup can be obtained, after restricting to invariant subspaces, factorizing through invariant subspaces and similarity transforms, from a bounded analytic semigroup on some bigger $L_p$-space which is positive and contractive on the real line.
The objective of this paper is to study wandering subspaces for commuting tuples of bounded operators on Hilbert spaces. It is shown that, for a large class of analytic functional Hilbert spaces $\mathcal{H}_K$ on the unit ball in $\mathbb C^n$, wandering subspaces for restrictions of the multiplication tuple $M_z = (M_{z_1}, \ldots ,M_{z_n})$ can be described in terms of suitable $\mathcal{H}_K$-inner functions. We prove that $\mathcal{H}_K$-inner functions are contractive multipliers and deduce a result on the multiplier norm of quasi-homogenous polynomials as an application. Along the way w…
Given a surjective mapping $f : E \to F$ between Banach spaces, we investigate the existence of a subspace $G$ of $E$, with the same density character as $F$, such that the restriction of $f$ to $G$ remains surjective. We obtain a positive answer whenever $f$ is continuous and uniformly open. In the smooth case, we deduce a positive answer when $f$ is a $C^1$-smooth surjection whose set of critical values is countable. Finally we show that, when $f$ takes values in the Euclidean space $\mathbb R^n$, in order to obtain this result it is not sufficient to assume that the set of critical values o…
Let T:=[T_1,..., T_n] be an n-tuple of operators on a Hilbert space such that T is a completely non-coisometric row contraction. We establish the existence of a "one-to-one" correspondence between the joint invariant subspaces under T_1,..., T_n, and the regular factorizations of the characteristic function associated with T. In particular, we prove that there is a non-trivial joint invariant subspace under the operators T_1,..., T_n, if and only if there is a non-trivial regular factorization of the characteristic function. We also provide a functional model for the joint invariant subspaces …
In 1965, Ron Douglas proved that if $X$ is a closed subspace of an $L^1$-space and $X$ is isometric to another $L^1$-space, then $X$ is the range of a contractive projection on the containing $L^1$-space. In 1977 Arazy-Friedman showed that if a subspace $X$ of $C_1$ is isometric to another $C_1$-space (possibly finite dimensional), then there is a contractive projection of $C_1$ onto $X$. In 1993 Kirchberg proved that if a subspace $X$ of the predual of a von Neumann algebra $M$ is isometric to the predual of another von Neumann algebra, then there is a contractive projection of the predual of…
