Algebras Intertwining Normal and Decomposable Operators

The question whether every operator on infinite-dimensional Hilbert space 𝐻 has a nontrivial invariant subspace or a nontrivial hyperinvariant subspace is one of the most difficult problems in operator theory. This problem is open for more than half a century. A subnormal operator has a nontrivial invariant subspace, but the existence of nontrivial invariant subspace for a hyponormal operator 𝑇 still open. In this paper we give an affirmative answer of the existence of a nontrivial hyperinvariant subspace for a hyponormal operator. More generally, we show that a large classes of operators containing the class of hyponormal operators have nontrivial hyperinvariant subspaces. Finally, every generalized scalar operator on a Banach space 𝑋 has a nontrivial invariant subspace.

We use the method of minimal vectors to prove that certain classes of positive quasinilpotent operators on Banach lattices have invariant subspaces. We say that a collection of operators F on a Banach lattice X satisfies condition ( ) if there exists a closed ball B(x0,r) in X such that x0 > 0 and kx0k > r, and for every sequence (xn) in B(x0,r) (0,x0) there exists a subsequence (xni) and a sequence Ki 2 F such that Kixni converges to a non-zero vector. Let Q be a positive quasinilpotent operator on X, one-to-one, with dense range. Denote hQ) = {T > 0 : TQ 6 QT}. If either the set of all operators dominated by Q or the set of all contractions in hQ) satisfies ( ), then hQ) has a common invariant subspace. We also show that if Q is a one-to-one quasinilpotent interval preserving operator on C0(), then hQ) has a common invariant subspace. Lomonosov proved in (Lom73) that if T is not a multiple of the identity and com- mutes with a non-zero compact operator K, then T has a hyperinvariant subspace, that is, a proper closed nontrivial subspace invariant under every operator S in the commutant {T} 0 = {S 2 L(X) : ST = TS}. There has been numerous extensions and generalizations of the result of Lomonosov. In particular, Abramovich, Aliprantis, and Burkinshaw produced several generalizations of Lomonosov's theorem for Banach lattice setting (AAB93, AAB94, AAB98), see also (AA02). In these generalizations commutation relations are substituted by a super-commutation relation ST 6 TS or ST > TS and domination 0 6 K 6 T. They proved a series of results of the fol- lowing type: if S is related to a compact operator via a certain rather loose chain of super-commutations and dominations, then S has an invariant subspace. Ansari and Enflo (AE98) have recently introduced the so-called technique of minimal vectors in order to prove the existence of invariant subspaces for certain classes of operators on a Hilbert space. The method was later modified so that it could be used in arbitrary Banach spaces in (JKP03, And03, CPS04, Tr04). In particular, the method of minimal vectors allows to prove Lomonosov-type results where a compact operator is replaced with a family of operators that mimic a compact operator.

Over the past years, a significant interest has developed in the study of holomorphic functions defined on a domain in an infinite-dimensional Banach space and of their constituents (via Taylor expansions), the homogeneous polynomials. Many of the questions that have been studied have arisen from considerations of infinitedimensional topology and from standard function algebra questions [CCG]. Recently, there has been an interest in connecting the well-developed theory of the geometry of Banach spaces with the function theory questions that have been studied classically, and some progress has been made in this direction [ACG; D; F; CCG; CGJ]. In addition, connections between properties of polynomials and geometry of the unit ball has been of interest (see [GJL] for a survey of this topic). The present work is an attempt to study some of the properties of bounded analytic functions on the unit ball of an infinite-dimensional Banach space. In particular, we are interested in understanding something of boundary behavior; we combine techniques from the several fields to investigate it, especially with regard to the interplay with convexity and smoothness. Many of the results here apply to the classical “nice” reflexive spaces, such as lp and Lp (1 < p <∞). It is almost certain that there is much more to be learned even about the Hilbert space case. We consider the boundary behavior of H∞ functions on B, the open unit ball of an infinite-dimensional complex Banach space that has the geometric properties of uniform convexity, uniform smoothness, or both. By uniform smoothness, we mean uniform (real) Frechet differentiability of the norm, with the space considered as a real Banach space. Uniform convexity will mean that the dual is uniformly smooth; since spaces with either property are reflexive, this definition is complete. To be specific however, we state the following (after [LT]).

We show that the celebrated Lomonosov theorem cannot be improved by increasing the number of commuting operators. Specifically, we prove that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T colon script l 1 right-arrow script l 1"> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo> : </mml:mo> <mml:msub> <mml:mi> ℓ </mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo stretchy="false"> → </mml:mo> <mml:msub> <mml:mi> ℓ </mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">T\colon \ell _1\to \ell _1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the operator without a non-trivial closed invariant subspace constructed by C. J. Read, then there are three operators <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S 1"> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">S_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S 2"> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">S_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (non-multiples of the identity) such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> commutes with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S 1"> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">S_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S 1"> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">S_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> commutes with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S 2"> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">S_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S 2"> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">S_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> commutes with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is compact. It is also shown that the commutant of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> contains only series of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .

Following the scalar-valued case considered by Djakow and Ramanujan (A remark on Bohr’s theorem and its generalizations 14:175–178, 2000) we introduce, for each complex Banach space X and each \(1\le p<\infty \), the p-Bohr radius of X as the value $$\begin{aligned} r_p(X)= \sup \bigg \{r\ge 0: \sum _{n=0}^\infty \Vert x_n\Vert ^pr^{np}\le \sup _{|z|<1} \Vert f(z)\Vert ^p\bigg \} \end{aligned}$$ where \(x_n\in X\) for each \(n\in \mathbb {N}\cup \{0\}\) and \(f(z)=\sum _{n=0}^\infty x_nz^n\in H^\infty (\mathbb {D},X)\). We show that a complex (possibly infinite dimensional) Banach space X is p-uniformly \(\mathbb {C}\)-convex for \(p\ge 2\) if and only if \(r_{p}(X)>0\). We study the p-Bohr radius of the Lebesgue spaces \(L^q(\mu )\) for different values of p and q. In particular we show that \(r_p(L^q(\mu ))=0\) whenever \(p<2\) and \(dim(L^q(\mu ))\ge 2\) and \(r_p(L^q(\mu ))=1\) whenever \(p\ge 2\) and \(p'\le q\le p\). We also provide some lower estimates for \(r_2(L^q(\mu ))\) for the values \(1\le q<2\).

The multiplicities of the powers of a bounded linear operator T T , acting on a complex separable infinite-dimensional Banach space X \mathfrak {X} , satisfy the inequalities \[ ( ∗ ∗ ) μ ( T n ) ⩽ μ ( T h n ) ⩽ h μ ( T n ) for all h , n ⩾ 1. ( * * )\qquad \mu ({T^n}) \leqslant \mu ({T^{hn}}) \leqslant h\mu ({T^n})\quad {\text {for}}\;{\text {all}}\;h,n \geqslant 1. \] Nothing else can be said, in general, because simple examples show that for each sequence { μ n } n = 1 ∞ \{ {\mu _n}\} _{n = 1}^\infty , satisfying the inequalities ( ∗ ∗ ) ( * * ) , there exists T T acting on X \mathfrak {X} such that μ ( T n ) = μ n \mu ({T^n}) = {\mu _n} for all n ⩾ 1 n \geqslant 1 .

Hilden's simple proof of Lomonosov's invariant subspace theorem

Compact operators, invariant subspaces, and spectral synthesis

Let $${\mathbf {B}}_{{\mathbb {X}}}$$ be the open unit ball of a complex Banach space $${\mathbb {X}}$$ , which may beinfinite dimensional. The weighted composition operator and weighted space defined on $${\mathbf {B}}_{{\mathbb {X}}}$$ are introduced. We obtain the boundedness and compactness of the weightedcomposition operator from the Bloch-type spaces to the weighted spaces, and some properties with the Bloch-type spaces are given. Our main results generalize theprevious works on the Euclidean unit ball $${\mathbb {B}}^n$$ to the case of $${\mathbf {B}}_{{\mathbb {X}}}$$ .

The main point of this paper is to define the multiplicity of zero and to establish Rouche's theorem for holomorphic maps in infinite dimensional complex Banach spaces.

In this paper we introduce and study the M-hypercyclicity of strongly continuous cosine function on separable complex Banach space, and we give the criteria for cosine function to be M-hypercyclic. We also prove that every separable infinite dimensional complex Banach space admits a uniformly continuous cosine function.

Let X be a finite or infinite dimensional complex Banach space. We characterize the bounded weighted composition operators between the Bloch type space and the Hardy space on the unit ball of X, extending several known results for finite dimensional domains.

We study classes of mappings between finite and infinite dimensional Banach spaces that are monotone and mappings which are differences of monotone mappings (DM). We prove a Radó–Reichelderfer estimate for monotone mappings in finite dimensional spaces that remains valid for DM mappings. This provides an alternative proof of the Fréchet differentiability a.e. of DM mappings. We establish a Morrey-type estimate for the distributional derivative of monotone mappings. We prove that a locally DM mapping between finite dimensional spaces is also globally DM. We introduce and study a new class of the so-called UDM mappings between Banach spaces, which generalizes the concept of curves of finite variation.

In this paper, we shall generalize the unicellularity of operators on finite-dimensional spaces to that of the contraction of class Co on Hilbert spaces. We prove: (1) Each nilpotent operator on Hilbert space is Banach reducible (Theorem 3). (2) A contraction T of class CO on Hilbert space is unicellular if and only if T has one-point spectrum and every invariant subspace for T is cyclic (Theorem 6). (3) A contraction T of class Co on Hilbert space is unicellular if and only if T has one-point spectrum and all invariant subspaces of T are hyperinvariant subspaces of T (Theorem 8). The aim of this paper is to discuss the unicellularity of contractions of class Co. We shall generalize the unicellularity of operators acting on finitedimensional spaces to that on infinite-dimensional spaces. Throughout this paper, we denote by H a complex, separable, and infinitedimensional Hilbert space. The related concepts and symbols can be found in [4]. Definition 1. Let H1 be a nontrivial invariant subspace of T. H1 is called a Banach reducible subspace of T if there exists another nontrivial invariant subspace H2 of T such that

The author introduces the notion of an analytically decomposable operator which generalizes the decomposable operator due to C. Foias in that the spectral decompositions of the underlying Banach space (1) admit a wider class of invariant subspaces called “analytically invariant” and (2) span the space only densely. It is shown that analytic decomposability is stable under the functional calculus, direct sums and restrictions to certain kinds of invariant subspaces, as well as perturbation by commuting scalar operators. It is fundamental for many of these results that every analytically decomposable operator has the single-valued extension property. An extensive investigation of analytically invariant subspaces is given. The author shows by example that this class is distinct from those of spectral maximal and hyperinvariant subspaces, but he further shows that analytically invariant subspaces have many useful spectral properties. Some applications of the general theory are made. For example, it is shown that under certain restrictions an analytically decomposable operator is decomposable.