Cyclic Subnormal Operator Research Articles

Spectral analysis of 2D outlier layout

Thompson’s partition of a cyclic subnormal operator into normal and completely non-normal components is combined with a non-commutative calculus for hyponormal operators for separating outliers from the cloud, in rather general point distributions in the plane. The main result provides exact transformation formulas from the power moments of the prescribed point distribution into the moments of the uniform mass carried by the cloud. The proposed algorithm solely depends on the Hessenberg matrix associated to the original data. The robustness of the algorithm is reflected by the insensitivity of the output under trace class, or by a theorem of Voiculescu, under certain Hilbert–Schmidt class, additive perturbations of the Hessenberg matrix.

Spectral picture for rationally multicyclic subnormal operators

For a pure bounded rationally cyclic subnormal operator $S$ on a separable complex Hilbert space $\mathcal H,$ J. B. Conway and N. Elias (Analytic bounded point evaluations for spaces of rational functions, J. Functional Analysis, 117:1{24, 1993) showed that $clos(\sigma (S) \setminus \sigma_e (S)) = clos(Int (\sigma (S))).$ This paper examines the property for rationally multicyclic (N-cyclic) subnormal operators. We show: (1) There exists a 2-cyclic irreducible subnormal operator $S$ with $clos(\sigma (S) \setminus \sigma_e (S)) \neq clos(Int (\sigma (S))).$ (2) For a pure rationally $N-$cyclic subnormal operator $S$ on $\mathcal H$ with the minimal normal extension $M$ on $\mathcal K \supset \mathcal H,$ let $\mathcal K_m = clos (span\{(M^*)^kx: ~x\in\mathcal H,~0\le k \le m\}.$ Suppose $M |_{\mathcal K_{N-1}}$ is pure, then $clos(\sigma (S) \setminus \sigma_e (S)) = clos(Int (\sigma (S))).$

Subnormal operators whose adjoints have rich point spectrum

A generalized version of the Glauber–Klauder basic formula of quantum optics is shown to be valid for any cyclic subnormal operator S whose adjoint has a rich point spectrum σ p ( S ∗ ) (in the sense that a semispectral measure of S vanishes on C ∖ σ p ( S ∗ ) ∗ ). It is exhibited that such operators always have analytic models. The point spectrum of the adjoint of a subnormal operator which satisfies a generalized version of the Glauber–Klauder formula is proved to be rich (in the above sense).

Sectorial forms and unbounded subnormals

AbstractWe use the theory of sectorial sesquilinear forms to characterize the closure of the Creation Operator of Quantum Mechanics in the classical set-up. Further, we bring that theory to bear upon the class of unbounded cyclic subnormal operators that admit analytic models; in particular, we provide a sufficient condition for the existence of complete sets of eigenvectors for certain sectorial operators related to unbounded subnormals. The relevant theory is illustrated in the context of a class of analytic models of which the classical Segal–Bargmann space is a prototype. The framework of sectorial sesquilinear forms is also shown to be specially useful for treating questions related to the existence, uniqueness and stability of certain parabolic evolution equations naturally associated with such analytic models.

On the range kernel orthogonality and p-symmetric operators

Let H be a separable infinite dimensional complex Hilbert space, and let L(H) denote the algebra of all bounded linear operators on H . For given A ∈ L(H) , we define the derivation δA : L(H) −→ L(H) by δA(X) = AX − XA . In this paper we establish the orthogonality of the range R(δA) and the kernel ker(δA) of a derivation δA induced by a cyclic subnormal operator A , in the usual sense. We give a version of the Putnam Fuglede theorem. We establish a short proof of the principal result of F. Wenying and J. Guoxing in [10]. Relatad results for P-symmetric operators are also given. Mathematics subject classification (2000): 47B47, 47B10, 47A30.

A local lifting theorem for subnormal operators

Criteria for the existence of lifts of operators intertwining subnormal operators are established. The main result of the paper reduces lifting questions for general subnormal operators to questions about lifts of cyclic subnormal operators. It is shown that in general the existence of local lifts (i.e. those coming from cyclic parts) for a pair of subnormal operators does not imply the existence of a global lift. However this is the case when minimal normal extensions of subnormal operators in question are star-cyclic.

Embry truncated complex moment problem

Let T be a cyclic subnormal operator on a Hilbert space H with cyclic vector x 0 and let γ ij :=( T * i T j x 0, x 0), for any i,j∈ N∪{0} . The Bram–Halmos’ characterization for subnormality of T involved a moment matrix M( n). In a parallel approach, we construct a moment matrix E( n) corresponding to Embry’s characterization for subnormality of T. We discuss the relationship between M( n) and E( n) via the full moment problem. Next, given a collection of complex numbers γ≡{ γ ij } (0⩽ i+ j⩽2 n, | i− j|⩽ n) with γ 00>0 and γ ji= γ ̄ ij , we consider the truncated complex moment problem for γ; this entails finding a positive Borel measure μ supported in the complex plane C such that γ ij=∫ z ̄ iz j dμ(z) . We show that this moment problem can be solved when E( n)⩾0 and E( n) admits a flat extension E( n+ k), where k=1 when n is odd and k=2 when n is even.

The commutant of rationally cyclic subnormal operators and rational approximation

Let Ω denote the set of analytic bounded point evaluations forR q (K, μ). Assume that $$\bar \Omega = K$$ . In this paper, we first show that if Ω is a finitely connected domain and if the evaluation map fromR q (K, μ)пL ∞(μ) toH ∞(Ω) is surjective, then μ|∂Ω is absolutely continuous with respect to harmonic measure for Ω. This generalizes Olin and Yang's corresponding result for polynomials and the proof we present here is simpler. We also provide an example that shows this absolute continuity property fails in general when Ω is an infinitely connected domain. In the second part, we then offer a solution to a problem of Conway and Elias.

The Berger-Shaw theorem for cyclic subnormal operators

This work gives a sharp form of the Berger-Shaw Theorem for cyclic subnormal operators. That is, if S is a cyclic subnormal operator, then S has trace class self- commutator and the trace equals (1=…)Area(ae(S)iaee(S)). It also characterizes those functions f such that f(S) has trace class self-commutator and computes the trace as the Dirichlet integral of ^ f on the set of analytic bounded point evaluations for S. The technique used also gives some partial results for rationally cyclic subnormal operators.

A Subnormal Operator and its Dual

AbstractIt is shown that the essential spectrum of a cyclic, self-dual, subnormal operator is symmetric with respect to the real axis. The study of the structure of a cyclic, irreducible, self-dual, subnormal operator is reduced to the operator Sμ with bpeμ = D. Necessary and sufficient conditions for a cyclic subnormal operator Sμ with bpeμ = D to be self-dual are obtained under the additional assumption that the measure on the unit circle is log-integrable. Finally, an approach to a general cyclic, self-dual, subnormal operator is discussed.

Invariant subspaces for algebras of subnormal operators. II

We continue our study of hyperinvariant subspaces for rationally cyclic subnormal operators. We establish a connection between hyperinvariant subspaces and weak-star continuous point evaluations on the commutant.

A Structure Theorem for the Commutant of a Class of Cyclic Subnormal Operators

A Structure Theorem for the Commutant of a Class of Cyclic Subnormal Operators

A structure theorem for the commutant of a class of cyclic subnormal operators

An m m -measure is defined to be a measure μ \mu such that the analytic bounded point evaluations of P 2 ( μ ) {P^2}(\mu ) is the open unit disk D {\mathbf {D}} in the complex plane, and the weak* closure of the analytic polynomials in L ∞ ( μ ) {L^\infty }(\mu ) is the set of bounded analytic functions on D {\mathbf {D}} . A complete characterization of P 2 ( μ ) ∩ L ∞ ( μ ) {P^2}(\mu ) \cap {L^\infty }(\mu ) , the commutant of the cyclic subnormal operator of multiplication by z z on P 2 ( μ ) {P^2}(\mu ) , is then obtained.

Invariant subspaces for algebras of subnormal operators

Every rationally cyclic subnormal operator has a hyperinvariant subspace.

The uniform algebra associated with a cyclic subnormal operator

A uniform algebra is associated with each cyclic subnormal operator, and a map is defined from the spectrum of the operator into the maximal ideal space of the uniform algebra. Affirmative answers as to whether this map is continuous are given under reasonable hypothesis on the structure of the spectrum of the operator.

Quasisimilar Operators in the Commutant of a Cyclic Subnormal Operator

Quasisimilar Operators in the Commutant of a Cyclic Subnormal Operator

Quasisimilar operators in the commutant of a cyclic subnormal operator

Compactly supported positive regular Borel measures on the complex plane that share "certain" properties with normalized arclength measure on the boundary of the unit disk are called m m -measures. Let μ \mu be an m m -measure and let S μ {S_\mu } be the cyclic subnormal operator of multiplication by z z on the closure of the polynomials in L 2 ( μ ) {L^2}(\mu ) . Necessary and sufficient conditions for an operator in the commutant of S μ {S_\mu } to be quasisimilar to S μ {S_\mu } are investigated. In particular it is shown that if the Bergman shift and an operator in its commutant are quasisimilar, then they are unitarily equivalent.

Cellular-indecomposable subnormal operators

A bounded operatorT is called cellular-indecomposable ifL ∩M ≠ {0} wheneverL andM are nonzero invariant subspaces forT. We prove that a cyclic subnormal operator is cellular-indecomposable if and only if it is quasi-similar to an analytic Toeplitz operator whose symbol is a weak-star generator ofH ∞. This completes our previous work [5], [6].

On Quasisimilarity for Subnormal Operators, II

AbstractLet S be a subnormal operator and let be the weak-star closed algebra generated by S and 1. An example of an irreducible cyclic subnormal operator S is found such that there is a T in with S and T quasisimilar but not unitarily equivalent. However, if S is the unilateral shift, T ∈ and S and T are quasisimilar, then S ≅ T.

An integrality theorem for subnormal operators

In a recent paper we conjectured that the principal function of a cyclic subnormal operator T is a.e. equal to the negative of a characteristic function. We showed that this was true in a variety of cases - including the general arc length Swiss Cheese. Now we prove stronger results. The conjecture is a consequence of: The principal function of a subnormal operator with trace class self-commutator assumes a.e. nonpositive integer values. It is an interesting fact that this integrality is a basic geometric property of subnormal operators and is not associated with any smoothness or "thinness" of the essential spectrum of T. This result is actually a simple corollary of a much more basic fact: The mosaic of a subnormal operator with trace class self-commutator is projection valued a.e. We have long known that the mosaic is a complete unitary invariant for T. Thus, this theorem establishes a map z ↦ Range B(z) which associates a subspace of Hilbert space with almost every point of the plane; and this generalized bundle completely characterizes the subnormal operator T. If T is cyclic then its mosaic B(·) is a.e. either the zero operator or a rank one projection.