Invariant subspaces and localizable spectrum

LetT L(X) be a continuous linear operator on a complex Banach spaceX. We show thatT possesses non-trivial closed invariant subspaces if its localizable spectrum _loc(T) is thick in the sense of the Scott Brown theory. Since for quotients of decomposable operators the spectrum and the localizable spectrum coincide, it follows that each quasiaffine transformation of a Banach-space operator with Bishop's property () and thick spectrum has a non-trivial invariant subspace. In particular it follows that invariant-subspace results previously known for restrictions and quotients of decomposable operators are preserved under quasisimilarity.     

Cyclicity results for Jordan and compressed Toeplitz operators

A vectorx in a Hilbert spaceH iscyclic for a bounded linear operatorTHH if the closed linear span of the orbit {T ⁿ xn0} ofx underT is all ofH. Operators which have a cyclic vector are said to be cyclic.Jordan operators are the infinite direct sums of Jordan cells acting on finite- dimensional Hilbert spaces. Necessary and sufficient conditions for a Jordan operator to be cyclic are given (see Corollary 6). In this case, a dense set of cyclic vectors is exhibited (see Corollary 4). Sufficient conditions for uncountable collections of cyclic Jordan operators to have a common cyclic vector are given and, in this case, a dense set of common cyclic vectors is exhibited (see Corollary 9).Analogues of these cyclicity results for Jordan operators are obtained for compressions of analytic Toeplitz operatorsT _A FAF on the Hardy spaceH ² to subspaces (BH ²) invariant for the backward shiftT _z ^* whereB is a Blaschke product by showing that such compressions are quasisimilar to Jordan operators.     

A note on operator norm inequalities

If P is a positive operator on a Hilbert space H whose range is dense, then a theorem of Foias, Ong, and Rosenthal says that: [(P)]^–1T[(P)]<-12 max {T, P^–1TP} for any bounded operator T on H, where is a continuous, concave, nonnegative, nondecreasing function on [0, P]. This inequality is extended to the class of normal operators with dense range to obtain the inequality [(N)]^–1T[(N)]<-12c² max {tT, N^–1TN} where is a complex valued function in a class of functions called vase-like, and c is a constant which is associated with by the definition of vase-like. As a corollary, it is shown that the reflexive lattice of operator ranges generated by the range NH of a normal operator N consists of the ranges of all operators of the form (N), where is vase-like. Similar results are obtained for scalar-type spectral operators on a Hilbert space.This author gratefully acknowledges the support of Central Michigan University in the form of a Research Professorship.     

Eigenfunctions of the hyperbolic composition operator

Letu inH ² be zero at one of the fixed points of a hyperbolic Möbius transform of the unit diskD. We will show, under some additional conditions onu, that the doubly cyclic subspaceS _u =V _n=– C ⁿ u contains nonconstant eigenfunctions of the composition operatorC . This implies that the cyclic subspace generated byu is not minimal. If there is an infinite dimensional minimal invariant subspace ofC (which is equivalent to the existance of an operator with only trivial invariant subspaces), then it is generated by a function with singularities at the fixed points of .     

Constructing one-parameter transformations on white noise functions in terms of equicontinuous generators

Let be a barreled locally convex space. A continuous operator on is called anequicontinuous generator if { ⁿ /n!;n=0,1,2,...} is an equicontinuous family of operators. For each equicontinuous generator a one-parameter group of operators is constructed by means of power series. There is a one-to-one correspondence between the equicontinuous generators and the locally equicontinuous holomorphic one-parameter groups of operators. If two equicontinuous generators ₁, ₂ satisfy [₁,₂]=₂ for some thena₁+b₂ is also an equicontinuous generator for anya, b. These general results are applied to a study of operators on white noise functions. In particular, a linear combination of the number operator and the Gross Laplacian, which are natural infinite dimensional analogues of a finite dimensional Laplacian, is always an equicontinuous generator. This result contributes to the Cauchy problems in white noise (Gaussian) space.Work supported by Alexander von Humboldt-Stiftung and Japan Society for Promotion of Sciences     

When do Toeplitz and Hankel operators commute?

We completely classify all Toeplitz and Hankel operators which commute; namely, we prove that that a non-trivial Hankel operator and a non-trivial Toeplitz operator commute if and only if the Hankel operator has symbolz, where is the symbol of the Toeplitz operator, and is an affine function of the characteristic function of certain anti-symmetric sets of the unit circle.     

Boundary-value problems for two-dimensional canonical systems

The two-dimensional canonical systemJy=–Hy where the nonnegative Hamiltonian matrix functionH(x) is trace-normed on (0, ) has been studied in a function-theoretic way by L. de Branges in [5]–[8]. We show that the Hamiltonian system induces a closed symmetric relation which can be reduced to a, not necessarily densely defined, symmetric operator by means of Kac' indivisible intervals; of. [33], [34]. The formal defect numbers related to the system are the defect numbers of this reduced minimal symmetric operator. By using de Branges' one-to-one correspondence between the class of Nevanlinna functions and such canonical systems we extend our canonical system from (0, ) to a trace-normed system on which is in the limit-point case at ±. This allows us to study all possible selfadjoint realizations of the original system by means of a boundaryvalue problem for the extended canonical system involving an interface condition at 0.     

Analysis of Non-normal Operators via Aluthge Transformation

Let T be a bounded linear operator on a complex Hilbert space . In this paper, we show that T has Bishops property () if and only if its Aluthge transformation has property (). As applications, we can obtain the following results. Every w-hyponormal operator has property (). Quasi-similar w-hyponormal operators have equal spectra and equal essential spectra. Moreover, in the last section, we consider Chs problem that whether the semi-hyponormality of T implies the spectral mapping theorem Re(T) = (Re T) or not.     

Topological structure of the space of composition operators onH ∞

Components and isolated points of the topological space of composition operators onH in the uniform operator topology are characterized. Compact differences of two composition operators are also characterized. With the aid of these results, we show that a component inC(H ) is not in general the set of all composition operators that differ from the given one by a compact operator.Supported by the Grant-in-Aid for Scientific Research (C), the Ministory of Education, Science and Culture, No. 09640218, and the Nippon Institute of Technology No. 111Supported by the Japan Society for the Promotion of Science     

Factoring positive operators on reproducing kernel Hilbert spaces

We characterize when positive operators can be factored by analytic Toeplitz type operators. As a corollary, we give an operator theory characterization of those invariant subspaces of doubly commuting unilateral shifts, which are generated by a single inner function on the bidisk. The last result extends to shifts of arbitrary (countable) multiplicity.     

Invariance of closed convex sets and domination criteria for semigroups

Let a and b be two positive continuous and closed sesquilinear forms on the Hilbert space H=L ²(, ). Denote by T=T(t) _t0and S=S(t) _t0the semigroups generated by a and b on H. We give criteria in terms of a and b guaranteeing that the semigroup T is dominated by S, i.e. |T(t)f|S(t)|f| for all t0 and fH. The method proposed uses ideas on invariance of closed convex sets of H under semigroups. Applications to elliptic operators and concrete examples are given.     

Scaling limit of anticommuting self-adjoint operators and applications to Dirac operators

LetA andB be two anticommuting self-adjoint operators andV() be a symmetric operator in a Hilbert space, where >0 is a parameter. It is proven that, under some conditions forV(), the resolvents of A+² B±²|B|+V() converge as . Applications to the nonrelativistic-limit problem of Dirac operators and supersymmetry are discussed.This work is supported by the Grant-In-Aid 0560139 for science research from the Ministry of Education, Japan.     

On semibounded restrictions of self-adjoint operators

After the von Neumann's remark [10] about pathologies of unbounded symmetric operators and an abstract theorem about stability domain [9], we develope here a general theory allowing to construct semibounded restrictions of selfadjoint operators in explicit form. We apply this theory to quantum-mechanical momentum (position) operator to describe corresponding stability domains. Generalization to the case of measurable functions of these operators is considered. In conclusion we discuss spectral properties of self-adjoint extensions of constructed self-adjoint restrictions.     

An operator version of the Newman-Shapiro isometry theorem

We establish an operator version of the Newman — Shapiro Isometry Theorem for operators satisfying generalized canonical commutation relations. An application to operator inequalities is also given.     

Chaotic unbounded differentiation operators

We construct dense sets of hypercyclic vectors for unbounded differention operators, including differentiation operators on the Hardy spaceH ₂, and the Laplacian operator onL ₂((), for any bounded open subset of ². Furthermore, we show that these operators are chaotic, in the sense of Devaney.     

On the decomposition theorem for meromorphic Fredholm resolvents

In this note results of B. Gramsch and W. Kaballo [8] on the decomposition of meromorphic (semi-) Fredholm resolvents are sharpened. A condition on an Orlicz function is given, under which the singular part in this decomposition can be chosen meromorphic inN , the ideal of -nuclear operators. Then the necessity of this condition is studied. Moreover, it is shown that for the rather steep Orlicz functions relevant to this question,N equalsS , the ideal of -approximable operators.Dedicated to Professor Albert Schneider on the occasion of his 60 ^th birthdayresearch supported by a grant from DAAD     

Quasi-coherence of Hardy spaces in several complex variables

In this paper, we prove that the Hardy spaceH ^p (), 1p<, over a strictly pseudoconvex domain in ⁿ with smooth boundary is quasi-coherent. More precisely, we show that Toeplitz tuplesT with suitable symbols onH ^p () have property (). This proof is based on a well known exactness result for the tangential Cauchy-Riemann complex.     

ω-Hyponormal operators

The class of -hyponormal operators is introduced. This class contains allp-hyponormal operators. Certain properties of this class of operators are obtained. Among other things, it is shown that ifT is -hyponormal, then its spectral radius and norm are identical, and the nonzero points of its joint point spectrum and point spectrum are identical. Conditions under which a -hyponormal operator becomes normal, self-adjoint and unitary are given.     

Bernstein functions, complete hyperexpansivity and subnormality—I

Special classes of functions on the classical semigroupN of non-negative integers, as defined using the classical backward and forward difference operators, get associated in a natural way with special classes of bounded linear operators on Hilbert spaces. In particular, the class of completely monotone functions, which is a subclass of the class of positive definite functions ofN, gets associated with subnormal operators, and the class of completely alternating functions, which is a subclass of the class of negative definite functions onN, with completely hyper-expansive operators. The interplay between the theories of completely monotone and completely alternating functions has previously been exploited to unravel some interesting connections between subnormals and completely hyperexpansive operators. For example, it is known that a completely hyperexpansive weighted shift with the weight sequence {_n}(n0) (of positive reals) gives rise to a subnormal weighted shift whose weight sequence is {1/_n}(n0). The present paper discovers some new connections between the two classes of operators by building upon some well-known results in the literature that relate positive and negative definite functions on cartesian products of arbitrary sets using Bernstein functions. In particular, it is observed that the weight sequence of a completely hyperexpansive weighted shift with the weight sequence {_n}(n0) (of positive reals) gives rise to a subnormal weighted shift whose weight sequence is {_n+1/_n}(n0). It is also established that the weight sequence of any completely hyperexpansive weighted shift is a Hausdorff moment sequence. Further, the connection of Bernstein functions with Stieltjes functions and generalizations thereof is exploited to link certain classes of subnormal weighted shifts to completely hyperexpansive ones.