On completely irreducible operators

Let T be a bounded linear operator on Hilbert space H, M an invariant subspace of T. If there exists another invariant subspace N of T such that H = M + N and M ∩ N = 0, then M is said to be a completely reduced subspace of T. If T has a nontrivial completely reduced subspace, then T is said to be completely reducible; otherwise T is said to be completely irreducible. In the present paper we briefly sum up works on completely irreducible operators that have been done by the Functional Analysis Seminar of Jilin University in the past ten years and more. The paper contains four sections. In section 1 the background of completely irreducible operators is given in detail. Section 2 shows which operator in some well-known classes of operators, for example, weighted shifts, Toeplitz operators, etc., is completely irreducible. In section 3 it is proved that every bounded linear operator on the Hilbert space can be approximated by the finite direct sum of completely irreducible operators. It is clear that a completely irreducible operator is a rather suitable analogue of Jordan blocks in L(H), the set of all bounded linear operators on Hilbert space H. In section 4 several questions concerning completely irreducible operators are discussed and it is shown that some properties of completely irreducible operators are different from properties of unicellular operators. __________ Translated from Acta Sci. Nat. Univ. Jilin, 1992, (4): 20–29     

A characteristic operator function for the class of n-hypercontractions

We consider a class of bounded linear operators on Hilbert space called n-hypercontractions which relates naturally to adjoint shift operators on certain vector-valued standard weighted Bergman spaces on the unit disc. In the context of n-hypercontractions in the class C0⋅ we introduce a counterpart to the so-called characteristic operator function for a contraction operator. This generalized characteristic operator function Wn,T is an operator-valued analytic function in the unit disc whose values are operators between two Hilbert spaces of defect type. Using an operator-valued function of the form Wn,T, we parametrize the wandering subspace for a general shift invariant subspace of the corresponding vector-valued standard weighted Bergman space. The operator-valued analytic function Wn,T is shown to act as a contractive multiplier from the Hardy space into the associated standard weighted Bergman space.     

Geometric quantization, complex structures and the coherent state transform

It is shown that the heat operator in the Hall coherent state transform for a compact Lie group K (J. Funct. Anal. 122 (1994) 103-151) is related with a Hermitian connection associated to a natural one-parameter family of complex structures on T^*K. The unitary parallel transport of this connection establishes the equivalence of (geometric) quantizations of T^*K for different choices of complex structures within the given family. In particular, these results establish a link between coherent state transforms for Lie groups and results of Hitchin (Comm. Math. Phys. 131 (1990) 347-380) and Axelrod et al. (J. Differential Geom. 33 (1991) 787-902).     

Homogeneous operators on Hilbert spaces of holomorphic functions

In this paper we construct a large class of multiplication operators on reproducing kernel Hilbert spaces which are homogeneous with respect to the action of the Möbius group consisting of bi-holomorphic automorphisms of the unit disc D. Indeed, this class consists of exactly those operators for which the associated unitary representation of the universal covering group of the Möbius group is multiplicity free. For every m∈N we have a family of operators depending on m+1 positive real parameters. The kernel function is calculated explicitly. It is proved that each of these operators is bounded, lies in the Cowen-Douglas class of D and is irreducible. These operators are shown to be mutually pairwise unitarily inequivalent.     

Control subspaces of minimal dimension. Elementary introduction. Discotheca

In this paper there is introduced and studied the following characteristic of a linear operator A acting on a Banach space Χ: , where Cyc A=R∶R is a subspace of Χ, dim R<+∞. Spqn (AⁿR∶n?0)=χ. Always disc A ?μ_A=(the multiplicity of the spectrum of the operator (dim R∶R∈Cyc A), where (by definition) in each A-cyclic subspace there is contained a cyclic subspace of dimension ? disc A. For a linear dynamical system x(t)=Ax(t)+Bu,(t) which is controllable, the characteristic disc A of the evolution operator A shows how much the control space can be diminished without losing controllability. In this paper there are established some general properties ofdisc (for example, conditions are given under which disc(A⊕B))=max(discA, disc B); disc is computed for the following operators: S (S is the shift in the Hardy space H²); disc S=2, (but μ_S=i); disc S _n ^* =n (butμ=1), where S_n=S⊕. ⊕S; disc S=2, (but μ_S=1), where S is the bilateral shift. It is proved that for a normal operator N with simple spectrum, disc N=μ_N=1 ? (the operator N is reductive). There are other results also, and also a list of unsolved problems.     

ON REFLEXIVITY OF HYPONORMAL AND WEIGHTED SHIFT OPERATORS

By an elementary proof, we use a result of Conway and Dudziak to show that if A is a hyponormal operator with spectral radius r(A) such that its spectrum is the closed disc {z:|z| ≤ r(A)} then A is reflexive. Using this result, we give a simple proof of a result of Bercovici, Foias, and Pearcy on reflexivity of shift operators. Also, it is shown that every power of an invertible bilateral weighted shift is reflexive.     

The spectral picture and the closure of the similarity orbit of strongly irreducible operators

An operator on a complex, separable, infinite dimensional Hilbert space is strongly irreducible if it does not commute with any nontrivial idempotent. This article answers the following questions of D. A. Herrero: (i) Given an operatorT with connected spectrum, can we find a strongly irreducible operatorL such that they have same spectral picture? (ii) When we use a sequence of irreducible operators to approximateT, can the approximation be the “most economic”? i.e., does there exist a strongly irreducible operatorL such thatT ∈S(L) ^? (the closure of the similarity orbit ofL)? It is shown that the answer for the two questions is yes.     

Invertible composition operators on Hp

The weakly closed algebras generated by certain sets of composition operators are shown to be reflexive. A structure theorem for invertible composition operators on H² is obtained and used to show that such operators are reflexive. The structure theorem shows that invertible hyperbolic composition operators are similar to cosubnormal operators built up from bilateral weighted shifts. Another consequence of the structure theorem is that the composition operators induced by hyperbolic disc automorphisms are universal. Thus the general invariant subspace problem for Hilbert space operators is contained in the problem of determining the invariant subspace lattices of these operators.     

Spectral automorphisms in CB-effect algebras

Spectral automorphisms have been introduced in [IVANOV, A.—CARAGHEORGHEOPOL, D.: Spectral automorphisms in quantum logics, Internat. J. Theoret. Phys. 49 (2010), 3146–3152]_in an attempt to construct, in the abstract framework of orthomodular lattices, an analogue of the spectral theory in Hilbert spaces. We generalize spectral automorphisms to the framework of effect algebras with compression bases and study their properties. Characterizations of spectral automorphisms as well as necessary conditions for an automorphism to be spectral are given. An example of a spectral automorphism on the standard effect algebra of a finite-dimensional Hilbert space is discussed and the consequences of spectrality of an automorphism for the unitary Hilbert space operator that generates it are shown. The last section is devoted to spectral families of automorphisms and their properties, culminating with the formulation and proof of a Stone type theorem (in the sense of Stone’s theorem on strongly continuous one-parameter unitary groups — see, e.g. [REED, M.#x2014;SIMON, B.: Methods of Modern Mathematical Physics, Vol. I, Acad. Press, New York, 1975]) for a group of spectral automorphisms.     

p-Quasihyponormal operators have scalar extensions of order 6

In this paper we show that every p-quasihyponormal operator has a scalar extension of order 6, i.e., is similar to the restriction to a closed invariant subspace of a scalar operator of order 6, where 0<p<1. As a corollary, we get that every p-quasihyponormal operator with rich spectra has a nontrivial invariant subspace. Also we show that Aluthge transforms preserve an analogue of the single-valued extension property for W²(D,H) and an operator T.     

On operators commuting with Toeplitz operators modulo the compact operators

We prove that an operator on H² of the disc commutes modulo the compacts with all analytic Toeplitz operators if and only if it is a compact perturbation of a Toeplitz operator with symbol in H^∞ + C. Consequently, the essential commutant of the whole Toeplitz algebra is the algebra of Toeplitz operators with symbol in QC. The image in the Calkin algebra of the Toeplitz operators with symbol in H^∞ + C is a maximal abelian algebra. These results lead to a characterization of automorphisms of the algebra of compact perturbations of the analytic Toeplitz operators.     

Li–Yorke chaos translation set for linear operators

In order to study Li–Yorke chaos by the scalar perturbation for a given bounded linear operator T on a Banach space X, we introduce the Li–Yorke chaos translation set of T, which is defined by \(S_{LY}(T)=\{\lambda \in {\mathbb {C}};\lambda +T \text { is Li--Yorke chaotic}\}\). In this paper, some operator classes are considered, such as normal operators, compact operators, shift operators, and so on. In particular, we show that the Li–Yorke chaos translation set of the Kalisch operator on the Hilbert space \(\mathcal {L}^2[0,2\pi ]\) is a simple point set \(\{0\}\).     

Bifurcation for variational problems when the linearisation has no eigenvalues

For a bounded analytic function, ?, on the unit disk, $\text{D,}\text{let}\text{T}{}_{\mathrm{?}}\text{and}\text{M}{}_{\mathrm{?}}$ denote the operators of multiplication by ? on H²(?D) and L²(?D), respectively. In their 1973 paper, Deddens and Wong asked whether there is an analytic Toeplitz operator $\text{T}{}_{\mathrm{?}}$ that commutes with a nonzero compact operator, and whether every operator that commutes with an analytic Toeplitz operator has an extension that commutes with the corresponding multiplication operator on L². In the first part of this paper, we give an explicit example of an analytic Toeplitz operator T_φ that settles both of these questions. This operator commutes with a nonzero compact operator (a composition operator followed by an analytic Toeplitz operator). The only operators in the commutant of T_φ that extend to commute with M_φ are analytic Toeplitz operators. Although the commutant of T_φ contains more than just analytic Toeplitz operators, T_φ is irreducible. The remainder of the paper seeks to explain more fully the phenomena incorporated in this example by introducing a class of analytic functions, including the function φ, and giving additional conditions on functions g in the class to determine whether T_g commutes with nonzero compact operators, whether T_g is irreducible, and which operators in the commutant of T_g extend to the commutant of M_g. In particular, we find representations for operators in the commutant and second commutant of T_g.     

Strongly irreducible operators on Banach spaces

This paper firstly discusses the existence of strongly irreducible operators on Banach spaces. It shows that there exist strongly irreducible operators on Banach spaces with w*-separable dual. It also gives some properties of strongly irreducible operators on Banach spaces. In particular, if T is a strongly irreducible operator on an infinite-dimensional Banach space, then T is not of finite rank and T is not an algebraic operator. On Banach spaces with subsymmetric bases, including infinite-dimensional separable Hilbert spaces, it shows that quasisimilarity does not preserve strong irreducibility. In addition, we show that the strong irreducibility of an operator does not imply the strong irreducibility of its conjugate operator, which is not the same as the property in Hilbert spaces.     

On the automorphism group of a cone

Let V be a real finite dimensional vector space, and let C be a full cone in C. In Sec. 3 we show that the group of automorphisms of a compact convex subset of V is compact in the uniform topology, and relate the group of automorphisms of C to the group of automorphisms of a compact convex cross-section of C. This section concludes with an application which generalizes the result that a proper Lorentz transformation has an eigenvector in the light cone. In Sec. 4 we relate the automorphism group of C to that of its irreducible components. In Sec. 5 we show that every compact group of automorphisms of C leaves a compact convex cross-section invariant. This result is applied to show that if C is a full polyhedral cone, then the automorphism group of C is the semidirect product of the (finite) automorphism group of a polytopal cross-section and a vector group whose dimension is equal to the number of irreducible components of C. An example shows that no such result holds for more general cones.     

Shift-type invariant subspaces of contractions

Using the Sz.-Nagy-Foias functional model it was shown in [L. Kérchy, Injection of unilateral shifts into contractions with non-vanishing unitary asymptotes, Acta Sci. Math. (Szeged) 61 (1995) 443-476] that under certain conditions on a contraction T the natural embedding of a Hardy space of vector-valued functions into the corresponding L² space can be factored into the product of two transformations, intertwining T with a unilateral shift and with an absolutely continuous unitary operator, respectively. The norm estimates in the Factorization Theorem of this paper are sharpened to their best possible form by essential improvements in the proof. As a consequence we obtain that if the residual set of a contraction covers the whole unit circle then those invariant subspaces, where the restriction is similar to the unilateral shift with a similarity constant arbitrarily close to 1, span the whole space. Furthermore, the hyperinvariant subspace problem for asymptotically non-vanishing contractions is reduced to these special circumstances.     

Holomorphic Cartan geometry on manifolds with numerically effective tangent bundle

Let X be a compact connected Kähler manifold such that the holomorphic tangent bundle TX is numerically effective. A theorem of Demailly et al. (1994) [11] says that there is a finite unramified Galois covering M→X, a complex torus T, and a holomorphic surjective submersion f:M→T, such that the fibers of f are Fano manifolds with numerically effective tangent bundle. A conjecture of Campana and Peternell says that the fibers of f are rational and homogeneous. Assume that X admits a holomorphic Cartan geometry. We prove that the fibers of f are rational homogeneous varieties. We also prove that the holomorphic principal G-bundle over T given by f, where G is the group of all holomorphic automorphisms of a fiber, admits a flat holomorphic connection.     

Shift-Type Properties of Commuting,Completely Non Doubly Commuting Pairs of Isometries

Pairs (V, V′) of commuting, completely non doubly commuting isometries are studied. We show, that the space of the minimal unitary extension of V (denoted by U) is a closed linear span of subspaces reducing U to bilateral shifts. Moreover, the restriction of V′ to the maximal subspace reducing V to a unitary operator is a unilateral shift. We also get a new hyperreducing decomposition of a single isometry with respect to its wandering vectors which strongly corresponds with Lebesgue decomposition.     

<Emphasis Type="Italic">C</Emphasis>*-Algebras Generated by Mappings. Classification of Invariant Subspaces

We continue the study of an operator algebra associated with a self-mapping ? on a countable setX which can be represented as a directed graph. This C*-algebra belongs to a class of operator algebras, generated by a family of partial isometries satisfying some relations on their source and range projections. Earlier we have formulated the irreducibility criterion of such algebras, which give us a possibility to examine the structure of the corresponding Hilbert space. We will show that for reducible algebras the underlying Hilbert space can be represented either as an infinite sum of invariant subspaces or as a tensor product of a finite-dimensional Hilbert space with l²(Z). In the first case we present a conditions under which the studied algebra has an irreducible representation into a C*-algebra generated by a weighted shift operator. In the second case, the algebra has the irreducible finite-dimensional representations indexed by the unit circle.