Finite Defect Research Articles

Quasi-Similarity of Weak Contractions

Let T be a completely nonunitary (c.n.u.) weak contraction (in the sense of Sz.-Nagy and Foias). We show that T is quasi-similar to the direct sum of its CO part and C11 part. As a corollary, two c.n.u. weak contractions are quasi-similar to each other if and only if their CO parts and C11 parts are quasi-similar to each other, respectively. We also completely determine when c.n.u. weak contractions and CO contractions are quasi-similar to normal operators. Recall that a contraction T on the Hilbert space H is called a weak contraction if its spectrum a(T) does not fill the open unit disc D and 1 T* T is of finite trace. Contained in this class are all contractions T with finite defect index dTdim rank(1 T* T)'/2 and with a(T) =# D (cf. [9, p. 323]). Assume that T is a weak contraction which is also completely nonunitary (c.n.u.), that is, T has no nontrivial reducing subspace on which T is a unitary operator. For such a contraction, Sz.-Nagy and Foias obtained a C0-Cl1 decomposition and then found a variety of invariant subspaces which furnish its spectral decomposition (cf. [9, Chapter VIII]). In this note we are going to supplement other interesting properties of such contractions. We show that a c.n.u. weak contraction is quasi-similar to the direct sum of its C0 part and Cl part. Although the proof is not difficult, some of its interesting applications justify the elaboration here. An immediate corollary is that two such contractions are quasi-similar to each other if and only if their C0 parts are quasi-similar and their Cl parts are quasi-similar to each other. This is, in turn, used to show that two quasi-similar weak contractions have equal spectra. Another interesting consequence is that a c.n.u. weak contraction is quasi-similar to a normal operator if and only if its C0 part is. The latter can be shown to be equivalent to the condition that its minimal function is a Blaschke product with simple zeros, thus completely settling the question when a c.n.u. weak contraction is quasi-similar to a normal operator. Before we start to prove our main theorem, we provide some background work for our notations and terminology. The main reference is [9]. Presented to the Society September 9, 1976; received by the editors October 1, 1976. AMS (MOS) subject classifications (1970). Primary 47A45.

The hyperinvariant subspace lattice of a contraction of class 𝐶.₀

It is shown that if T is a C ⋅ 0 {C_{ \cdot 0}} contraction with finite defect indices, then Hyperlat T is (lattice) generated by those subspaces which are either ker ⁡ ψ ( T ) \ker \psi (T) or ran ran ξ ( T ) ¯ \overline {{\text {ran}}\;\xi (T)} , where ψ \psi and ξ \xi are scalar-valued inner functions.

Double commutants of 𝐶_{⋅0} contractions

D. Sarason has shown that an operator in the double commutant of a contraction T of class C 0 ( 1 ) {C_0}(1) is interpolated by a function in H ∞ {H^\infty } , that is, { T } ′ = { ϕ ( T ) ; ϕ ∈ H ∞ } \{ T\} ’ = \{ \phi (T);\phi \in {H^\infty }\} , [3]. Generally, in [4] Sz.-Nagy and C. Foiaş have shown that an operator in the double commutant of a contraction T of class C 0 ( n ) {C_0}(n) is interpolated by a function in N T {N_T} , that is, { T } = { ϕ 1 ( T ) − 1 ϕ 2 ( T ) : ϕ 2 / ϕ 1 ∈ N T } \{ T\} = \{ {\phi _1}{(T)^{ - 1}}{\phi _2}(T):{\phi _2}/{\phi _1} \in {N_T}\} . In this note we shall show that an operator in the double commutant of a C ⋅ 0 {C_{ \cdot 0}} contraction T with finite defect indices δ T ∗ > δ T {\delta _{{T^ \ast }}} > {\delta _T} is interpolated by a function in H ∞ {H^\infty } .

ON THE COMPLETENESS OF DERIVED CHAINS

We study the problem of completeness of the system of eigenvectors and associated vectors of operator-valued functions which are analytic in an angular region and which assume values in the ring of bounded linear operators in a separable Hilbert space . As a corollary of the fundamental theorem proved in this paper we obtain the following result. Theorem 1. Let , where 0$ SRC=http://ej.iop.org/images/0025-5734/29/1/A04/tex_sm_3650_img4.gif/>, and is a completely continuous positive operator; moreover, let for some 0$ SRC=http://ej.iop.org/images/0025-5734/29/1/A04/tex_sm_3650_img8.gif/>. Then for every 0$ SRC=http://ej.iop.org/images/0025-5734/29/1/A04/tex_sm_3650_img9.gif/> the closed linear hull of the eigenvectors and associated vectors of (or ) which correspond to the eigenvalues lying in the angular region has finite defect in .Bibliography: 20 titles.

Extensions of symmetric operators and symmetric binary relations

Various classes of extensions of symmetric operators with equal (finite or infinite) defect numbers are described in terms of abstract boundary conditions. The dual problem of the description of extensions of a symmetric binary relation is also considered.

Transversals of infinite families

Let U be a family of sets having a finite number of infinte members. A proof is given for a theorem of Brualdi and Scrimger regarding the existence of a transversal of U , and this theorem is used to obtain necessary and sufficient conditions for U to have a partial transversal of prescribed finite defect. The dual of the Brualdi-Scrimger result is discussed.

Finite size lattice defect in anisotropic copper: Interaction with an extended edge dislocation

Finite size lattice defect in anisotropic copper: Interaction with an extended edge dislocation

THE HELLY PROBLEM AND BEST APPROXIMATION IN A SPACE OF CONTINUOUS FUNCTIONS

Equivalence is verified between the Helly problem in the theory of moments and the problem of best approximation by elements of subspaces of finite defect. The existence and uniqueness conditions for the solution of these problems in a space of continuous functions are investigated.