The main result of the paper is that a system of invariant subspaces of a (completely non-unitary) Hilbert space contraction $T$ with finite defects (rank$(I-T^*T)<\infty$, rank$(I-TT^*)<\infty$) is an unconditional basis (Riesz basis) if and only if it is uniformly minimal.
Results of such type are quite well known: for a system of eigenspaces of a contraction with defects $1-1$ it is simply the famous Carleson interpolation theorem. For general invariant subspaces of operators with defects $1-1$ such theorem was proved by V. I. Vasyunin. Then partial results for the case of finite defects were obtained by the author.
The present paper solves the problem completely.
In this note we give a simple proof that every subspace of L_p, 2<p<infinity, with an unconditional basis has an equivalent norm determined by partitions and weights. Consequently L_p has a norm determined by partitions and weights.
We prove that the dual or any quotient of a separable reflexive Banach space with the unconditional tree property has the unconditional tree property. Then we prove that a separable reflexive Banach space with the unconditional tree property embeds into a reflexive Banach space with an unconditional basis. This solves several long standing open problems. In particular, it yields that a quotient of a reflexive Banach space with an unconditional finite dimensional decomposition embeds into a reflexive Banach space with an unconditional basis.
A generalization of Lozanovskii's result is proved. Let E be $k$-dimensional subspace of an $n$-dimensional Banach space with unconditional basis. Then there exist $x_1,..,x_k \subset E$ such that $B_E \p \subset \p absconv\{x_1,..,x_k\}$ and \[ \kla \frac{{\rm vol}(absconv\{x_1,..,x_k\})}{{\rm vol}(B_E)} \mer^{\frac{1}{k}} \kl \kla e\p \frac{n}{k} \mer^2 \pl .\] This answers a question of V. Milman which appeared during a GAFA seminar talk about the hyperplane problem. We add logarithmical estimates concerning the hyperplane conjecture for proportional subspaces and quotients of Banach spaces…
We prove that every lattice homomorphism acting on a Banach space $\mathcal{X}$ with the lattice structure given by an unconditional basis has a non-trivial closed invariant subspace. In fact, it has a non-trivial closed invariant ideal, which is no longer true for every positive operator on such a space. Motivated by these later examples, we characterize tridiagonal positive operators without non-trivial closed invariant ideals on $\mathcal{X}$ extending to this context a result of Grivaux on the existence of non-trivial closed invariant subspaces for tridiagonal operators.
For a large class of Banach spaces, a general construction of subspaces without local unconditional structure is presented. As an application it is shown that every Banach space of finite cotype contains either $l_2$ or a subspace without unconditional basis, which admits a Schauder basis. Some other interesting applications and corollaries follow.
We prove that if $X$ is a complex strictly monotone sequence space with $1$-unconditional basis, $Y \subseteq X$ has no bands isometric to $\ell_2^2$ and $Y$ is the range of norm-one projection from $X$, then $Y$ is a closed linear span a family of mutually disjoint vectors in $X$. We completely characterize $1$-complemented subspaces and norm-one projections in complex spaces $\ell_p(\ell_q)$ for $1 \leq p, q < \infty$. Finally we give a full description of the subspaces that are spanned by a family of disjointly supported vectors and which are $1$-complemented in (real or complex) Orlicz or …
We investigate different possiblities of subspaces of the space $\ell_{\infty}$ in terms of whether the subspaces are polyhedral or not. We further study finite-dimensional subspaces of $\ell_{\infty}$ which are of the form $\ell_\infty^n$ form some $ n \geq 2.$ As an application of the results we compute the number of extreme contractions for a class of the space of bounded linear operators. In particular we find the number of extreme contractions of $\mathbb{L}(\mathbb{X}, \ell_{\infty}^n),$ where $\mathbb{X}$ is a finite-dimensional polyhedral space.
Given a Polish topology $τ$ on ${\mathcal{B}_{1}(X)}$, the set of all contraction operators on $X=\ell_p$, $1\le p<\infty$ or $X=c_0$, we prove several results related to the following question: does a typical $T\in {\mathcal{B}_{1}(X)}$ in the Baire Category sense has a non-trivial invariant subspace? In other words, is there a dense $G_δ$ set $\mathcal G\subseteq ({\mathcal{B}_{1}(X)},τ)$ such that every $T\in\mathcal G$ has a non-trivial invariant subspace? We mostly focus on the Strong Operator Topology and the Strong$^*$ Operator Topology.
We give precise conditions under which the real interpolation space [Y_0,X_1]_{s,p} coincides with a closed subspace of the corresponding interpolation space [X_0,X_1]_{s,p} when Y_0 is a closed subspace of X_0 of codimension one. This result is applied to study the basis properties of nonharmonic Fourier series in Sobolev spaces H^s on an interval when 0<s<1. The main result: let E be a family of exponentials exp(i λ_n t) and E forms an unconditional basis in L^2 on an interval. Then there exist two number s_0, s_1 such that E forms an unconditional basis in H^s for s<s_0, E forms an uncondit…
We prove a general result on complemented unconditional basic sequences in Banach lattices and apply it to give some new examples of spaces with unique unconditional basis. We show that Tsirelson space and certain Nakano spaces have the unique unconditional bases. We also construct an example of a space with a unique unconditional basis with a complemented subspace failing to have a unique unconditional basis.
Using the Sz.-Nagy--Foias theory of contractions, we obtain general results about reducibility for a class of completely nonunitary contractions. These are applied to certain truncated Toeplitz operators, previously considered by Li--Yang--Lu and Gu. In particular, a negative answer is given to a conjecture stated by the latter.
If a Banach space is saturated with basic sequences whose linear span embeds into the linear span of any subsequence, then it contains a minimal subspace. It follows that any Banach space is either ergodic or contains a minimal subspace. For a Banach space $X$ with an (unconditional) basis, topological 0-1 law type dichotomies are stated for block-subspaces of $X$ as well as for subspaces of $X$ with a successive FDD on its basis. A uniformity principle for properties of block-sequences, results about block-homogeneity, and a possible method to construct a Banach space with an unconditional ba…
We show that any Banach space contains a continuum of non isomorphic subspaces or a minimal subspace. We define an ergodic Banach space $X$ as a space such that $E_0$ Borel reduces to isomorphism on the set of subspaces of $X$, and show that every Banach space is either ergodic or contains a subspace with an unconditional basis $ which is complementably universal for the family of its block-subspaces. We also use our methods to get uniformity results; for example, in combination with a result of B. Maurey, V. Milman and N. Tomczak-Jaegermann, we show that an unconditional basis of a Banach spa…
A commuting triple of Hilbert space operators $(A,B,P)$, for which the closed tetrablock $\overline{\mathbb E}$ is a spectral set, is called a \textit{tetrablock-contraction} or simply an $\mathbb E$-\textit{contraction}, where \[ \mathbb E=\{(x_1,x_2,x_3)\in \mathbb C^3:\, 1-x_1z-x_2w+x_3zw \neq 0 \quad \text{ whenever } \; |z|\leq 1, \; \; |w|\leq 1 \} \subset \mathbb C^3, \] is a polynomially convex domain which is naturally associated with the $μ$-synthesis problem. By applications of the theory of $\mathbb E$-contractions, we obtain several results on decompositions of contractions.
It is known that pure row contractions with one-dimensional defect spaces can be classified up to unitary equivalence by compressions of the standard $d$-shift acting on the full Fock space. Upon settling for a softer relation than unitary equivalence, we relax the defect condition and simply require the row contraction to admit a cyclic vector. We show that cyclic pure row contractions can be "transformed" (in a precise technical sense) into compressions of the standard $d$-shift. Cyclic decompositions of the underlying Hilbert spaces are the natural tool to extend this fact to higher multipl…
Let $X$ be a Banach space with an unconditional finite-dimensional Schauder decomposition $(E_n)$. We consider the general problem of characterizing conditions under which one can construct an unconditional basis for $X$ by forming an unconditional basis for each $E_n.$ For example, we show that if $\sup \dim E_n<\infty$ and $X$ has Gordon-Lewis local unconditional structure then $X$ has an unconditional basis of this type. We also give an example of a non-Hilbertian space $X$ with the property that whenever $Y$ is a closed subspace of $X$ with a UFDD $(E_n)$ such that $\sup\dim E_n<\infty$ th…
Using the tools of Sz.-Nagy--Foias theory of contractions, we describe in detail the invariant subspaces of the operator $ S\oplus S^* $, where $ S $ is the unilateral shift on a Hilbert space. This answers a question of Câmara and Ross.
The goal of the present paper is to push Sz.-Nagy--Foias model theory for a completely nonunitary Hilbert-space contraction operator $T$, to the case of a commuting pair of contraction operators $(T_1, T_2)$ having product $T = T_1 T_2$ which is completely nonunitary. The idea is to use the Sz.-Nagy-Foias functional model for $T$ as the model space also for the commutative tuple ($T_1, T_2)$ with $T = T_1 T_2$ equal to the usual Sz.-Nagy--Foias model operator, and identify what added structure is required to classify such commutative contractive factorizations $T = T_1 T_2$ up to unitary equiv…
Consider a finite collection $\{T_1, \ldots, T_J\}$ of differential operators with constant coefficients on $\mathbb{T}^2$ and the space of smooth functions generated by this collection, namely, the space of functions $f$ such that $T_j f \in C(\mathbb{T}^2)$. We prove that under a certain natural condition this space is not isomorphic to a quotient of a $C(S)$-space and does not have a local unconditional structure. This fact generalizes the previously known result that such spaces are not isomorphic to a complemented subspace of $C(S)$.
We present a Banach space $\mathfrak X$ with a Schauder basis of length $ω\_1$ which is saturated by copies of $c\_0$ and such that for every closed decomposition of a closed subspace $X=X\_0\oplus X\_1$, either $X\_0$ or $X\_1$ has to be separable. This can be considered as the non-separable counterpart of the notion of hereditarily indecomposable space. Indeed, the subspaces of $\mathfrak X$ have ``few operators'' in the sense that every bounded operator $T:X \to \mathfrak{X}$ from a subspace $X$ of $\mathfrak{X}$ into $\mathfrak{X}$ is the sum of a multiple of the inclusion and a $ω\_1$-sin…
If $E=\{e_i\}$ and $F=\{f_i\}$ are two 1-unconditional basic sequences in $L_1$ with $E$ $r$-concave and $F$ $p$-convex, for some $1\le r<p\le 2$, then the space of matrices $\{a_{i,j}\}$ with norm $\|\{a_{i,j}\}\|_{E(F)}=\big\|\sum_k \|\sum_l a_{k,l}f_l\|e_k\big\|$ embeds into $L_1$. This generalizes a recent result of Prochno and Schütt.
Assume that $T$ is a self-adjoint operator on a Hilbert space $\mathcal{H}$ and that the spectrum of $T$ is confined in the union $\bigcup_{j\in J}Δ_j$, $J\subseteq\mathbb{Z}$, of segments $Δ_j=[α_j, β_j]\subset\mathbb{R}$ such that $α_{j+1}>β_j$ and $$ \inf_{j} \left(α_{j+1}-β_j\right) = d > 0. $$ If $B$ is a bounded (in general non-self-adjoint) perturbation of $T$ with $\|B\|=:b<d/2$ then the spectrum of the perturbed operator $A=T+B$ lies in the union $\bigcup_{j\in J} U_{b}(Δ_j)$ of the mutually disjoint closed $b$-neighborhoods $U_{b}(Δ_j)$ of the segments $Δ_j$ in $\mathbb{C}$. Let $Q_j…
A particular case of results from [K2] is as follows. Let the unitary asymptote of a contraction $T$ contain the bilateral shift (of finite or infinite multiplicity). Then there exists an invariant subspace $\mathcal M$ of $T$ such that $T|_{\mathcal M}$ is similar to the unilateral shift of the same multiplicity. The proof is based on the Sz.-Nagy--Foias functional model for contractions. In the present paper this result is generalized to polynomially bounded operators, but in the simplest case. Namely, it is proved that if the unitary asymptote of a polynomially bounded operator $T$ contains…
In the first part of our paper, we show that $\ell_\infty$ has a dense linear subspace which admits an equivalent real analytic norm. As a corollary, every separable Banach space, as well as $\ell_1(\mathfrak{c})$, also has a dense linear subspace which admits an analytic renorming. By contrast, no dense subspace of $c_0(ω_1)$ admits an analytic norm. In the second part, we prove (solving in particular an open problem of Guirao, Montesinos, and Zizler) that every Banach space with a long unconditional Schauder basis contains a dense subspace that admits a $C^{\infty}$-smooth norm. Finally, we …
We construct for each $0<p\le 1$ an infinite collection of subspaces of $\ell_p$ that extend the example from [J. Lindenstrauss, On a certain subspace of $\ell_{1}$, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 12 (1964), 539-542] of a subspace of $\ell_{1}$ with no unconditional basis. The structure of this new class of $p$-Banach spaces is analyzed and some applications to the general theory of $\mathcal{L}_{p}$-spaces for $0<p<1$ are provided. The introduction of these spaces serves the purpose to develop the theory of conditional quasi-greedy bases in $p$-Banach spaces for $p<1$…
The paper is devoted to two particular cases of the following general problem. Let $α$ and $β$ be two types of bases in Banach spaces. Let a Banach space $X$ has bases of both types and a subspace $M\subset X^*$ contains the sequence of biorthogonal functionals of some $α$-basis in $X$. Does $M$ contain a sequence of biorthogonal functionals of some $β$-basis in $X$? The following particular cases are considered: $(α, β)$=(Schauder bases, unconditional bases), $(α, β)$=(Nonlinear operational bases, linear operational bases). The paper contains an investigation of some of the spaces constructed…
We show for $2\le p<\infty$ and subspaces $X$ of quotients of $L_{p}$ with a $1$-unconditional finite-dimensional Schauder decomposition that $K(X,\ell_{p})$ is an $M$-ideal in $L(X,\ell_{p})$.
A particular case of [07] was generalized from contractions to polynomially bounded operators in [G19]. Namely, it is proved in [G19] that if the unitary asymptote of a polynomially bounded operator $T$ contains the bilateral shift of multiplicity $1$, then there exists an invariant subspace $\mathcal M$ of $T$ such that $T|_{\mathcal M}$ is similar to the unilateral shift of multiplicity $1$. In the present paper, some corollaries of this result are given. In particular, reflexivity of polynomially bounded operators described above is proved.
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 …
If $X$ is a closed subspace of a Banach space $L$ which embeds into a Banach lattice not containing $\ell_\infty^n$'s uniformly and $L/X$ contains $\ell_\infty^n$'s uniformly, then $X$ cannot have local unconditional structure in the sense of Gordon-Lewis (GL-{\sl l.u.st.}).
We prove that a WLD subspace of the space $\ell_\infty^c(Γ)$ consisting of all bounded, countably supported functions on a set $Γ$ embeds isomorphically into $\ell_\infty$ if and only if it does not contain isometric copies of $c_0(ω_1)$. Moreover, a subspace of $\ell_\infty^c(ω_1)$ is constructed that has an unconditional basis, does not embed into $\ell_\infty$, and whose every weakly compact subset is separable (in particular, it cannot contain any isomorphic copies of $c_0(ω_1)$).
For graded Hilbert spaces $H$ and shift-like commuting tuples $T \in B(H)^n$, we show that each homogeneous joint invariant subspace $M$ of $T$ has finite index and is generated by its wandering subspace. Under suitable conditions on the grading $(H_k)_{k\geq 0}$ of $H$ the algebraic direct sum $\tilde{M} = \oplus_{k\geq 0} M \cap H_k$ becomes a finitely generated module over the polynomial ring $\mathbb C[z]$. We show that the wandering subspace $W_T(M)$ of $M$ is contained in $\tilde{M}$ and that each linear basis of $W_T(M)$ forms a minimal set of generators for the $\mathbb C[z]$-module $\…
Given Banach spaces E and F, we denote by ${\mathcal P}(^k!E,F)$ the space of all k-homogeneous (continuous) polynomials from E into F, and by ${\mathcal P}_{wb}(^k!E,F)$ the subspace of polynomials which are weak-to-norm continuous on bounded sets. It is shown that if E has an unconditional finite dimensional expansion of the identity, the following assertions are equivalent: (a) ${\mathcal P}(^k!E,F)={\mathcal P}_{wb}(^k!E,F)$; (b) ${\mathcal P}_{wb}(^k!E,F)$ contains no copy of $c_0$; (c) ${\mathcal P}(^k!E,F)$ contains no copy of $\ell_\infty$; (d) ${\mathcal P}_{wb}(^k!E,F)$ is complement…
We review how some multianalytic inner functions of the Beurling type theorem are associated to row contractions following works of G.Popescu. Motivated by a result on weak Markov dilations, we define a notion of characteristic function for ergodic and coisometric row contractions with a one-dimensional invariant subspace for the adjoints. Our characteristic function is a complete unitary invariant for such tuples. Thereafter we extend the analysis of characteristic functions to contractive liftings of row contractions. We apply our theory to completely positive maps and explore applications t…
Let T be a C_{\cdot 0}-contraction on a Hilbert space H and S be a non-trivial closed subspace of H. We prove that S is a T-invariant subspace of H if and only if there exists a Hilbert space D and a partially isometric operator Π: H^2_D(\mathbb{D}) \raro H such that ΠM_z = T Πand that S = ran Π, or equivalently, P_S = ΠΠ^*. As an application we completely classify the shift-invariant subspaces of C_{\cdot 0}-contractive and analytic reproducing kernel Hilbert spaces over the unit disc. Our results also includes the case of weighted Bergman spaces over the unit disk.
Let $1\le p <\infty$, $f\in L_p(\real)$ and $Λ\subseteq \real$. We consider the closed subspace of $L_p(\real)$, $X_p (f,Λ)$, generated by the set of translations $f_{(λ)}$ of $f$ by $λ\inΛ$. If $p=1$ and $\{f_{(λ)} :λ\inΛ\}$ is a bounded minimal system in $L_1(\real)$, we prove that $X_1 (f,Λ)$ embeds almost isometrically into $\ell_1$. If $\{f_{(λ)} :λ\inΛ\}$ is an unconditional basic sequence in $L_p(\real)$, then $\{f_{(λ)} : λ\inΛ\}$ is equivalent to the unit vector basis of $\ell_p$ for $1\le p\le 2$ and $X_p (f,Λ)$ embeds into $\ell_p$ if $2<p\le 4$. If $p>4$, there exists $f\in L_p(\re…
Invariant subspaces for finite index shifts in Hardy spaces
University of Rome Tor Vergata（罗马托尔维加塔大学）；Universidad Complutense de Madrid（马德里康普顿斯大学）；Instituto de Ciencias Matemáticas ICMAT(CSIC-UAM-UC3M-UCM)（数学科学研究所 ICMAT）
This paper is a follow-up contribution to our work [20] where we discussed some invariant subspace results for contractions on Hilbert spaces. Here we extend the results of [20] to the context of n-tuples of bounded linear operators on Hilbert spaces. Let T = (T_1, \ldots, T_n) be a pure commuting co-spherically contractive n-tuple of operators on a Hilbert space \mathcal{H} and \mathcal{S} be a non-trivial closed subspace of \mathcal{H}. One of our main results states that: \mathcal{S} is a joint T-invariant subspace if and only if there exists a partially isometric operator Π\in \mathcal{B}(…
The main issue we address in the present paper are the new models for completely non-unitary contractions with rank one defect operators acting on some Hilbert space of dimension $N\leq\infty$. This model complements nicely the well-known models of Liv$\rm{\check{s}}$ic and Sz.-Nagy--Foias. We show that each such an operator is unitarily equivalent to some truncated CMV matrix obtained from the ``full'' CMV matrix by deleting the first row and the first column, and acting in $\ell^2$ ($\dC^N$). This result can be viewed as a nonunitary version of the famous characterization of unitary operator…
