Nest Algebra And Related Topics

Throughout this paper, C, R, Q, Z, and N denote the set of complex numbers, the set of real numbers, the set of rational numbers, the set of integers, the set of natural numbers respectively. T and D denote the unit circle and the open unit disc in the complex plane respectively. H will always denote a complex separable infinite dimensional Hilbert space and B(H) denotes the set of all bounded linear operators on H.Nest algebras constitute the most important and tractable class of non-self-adjoint operator algebras on Hilbert space. The thesis deals with the nest algebra and related topics. It appears in three parts. In the first part of this thesis, we provide a brief introduction to nests and nest algebras and slant Toeplitz operators, review a few re-sults on connectedness. In the second part of this thesis, we consider the connectedness problem of invertibles in certain nest algebras. The last part of this thesis focuses on the subalgebra of a certain nest algebra which consists of generalized slant Toeplitz operators.A long-standing and well-known problem for nest algebra is the so called "con-nectedness problem", which asks whether the group of invertibles of a nest algebra is norm-pathwise connected. In the mid-1970s, Saeks raised the connectedness problem in his investigation of developing a Nyquist-like stability criterion for systems denned on a Hilbert resolution space. From then on, the connectedness problem have been promoted and considered by many authors. In spite of all the interest, there has been no significant progress until the emergence of the deep interpolation theorem of Orr. In 1993, Davidson and Orr achieved a breakthrough in solving the connectedness problem. Using the interpolation theorem of Orr, they showed that the invertibles are connected in each nest algebra of infinite multiplicity. This is the first significant progress about the connectedness problem. And soon after, in1994, Davidson, Orr and Pitts success-fully extended these results by showing that the invertibles are connected in a nest algebra provided there is a finite bound on the number of consecutive finite rank atoms in the nest. Their work reduces the connectedness problem for arbitrary nests to the case of upper triangular operators with respect to a fixed orthonormal basis.Let T be the unit circle in the complex plane with normalized Lesbegue measure. Let H=H2(T) be the usual Hardy space of all functions in L2(T) that have analytic extensions to the open unit disk D. Let W∈B(H) be the shift operator (Wf)(eiθ)=eiθf(eiθ). We will consider the nest N={{0}, H, WnH:n∈Z, n≥0} of subspaces of H, and its associated nest algebra AlgN={T E B(H):TWnH(?)WnH}. It should be mentioned that it was frequently conjectured that the connectedness prob-lem had a negative solution. Inspired by the fact that the invertible group of H∞(T) is not connected, Davidson provided a candidate for proving disconnectedness. Later Pitts proposed a function f which can't be connected to the constant function1via a norm continuous path within the group of invertible elements of the Banach algebra H∞(T), however the Toeplitz operator with symbol f can be connected to the identity via a norm continuous path of invertible elements in AlgN. Inspired by Pitts we shall show that a certain class invertible elements in AlgN can be connected to the identity in norm topology. The main results in Chapter2are the following theorems:Theorem1If T∈AlgN is invertible and σ(T) consists of finite points, then T can be connected to the identity through a path in the invertible group of AlgN.Theorem2Let h be an analytic function in the open unit disc D with bounded real part and h(n) denote the n-th. Fourier coefficient of h. If there exists a natural number k such that h(kl)=0for each l∈Z, then the Toeplitz operator Tf with symbol f can be connected to the identity through a path in the invertible group of AlgN, where f=eh.The study of slant Toeplitz operators originated in the practical applications. In the past few years, there have been intensive studies of wavelet transforms as an alternative for the Fourier transforms in many applications such as data compression or solution of certain types of differential equations. Several authors in this field have connected the smoothness of the wavelets with the spectral properties of the slant Toeplitz operators. They have not investigated basic properties of these operators but concentrated mainly on the applications to wavelets.In a1996paper Mark defined a slant Toeplitz operator as one whose j-th row below the base row is obtained by shifting the base row2j units to the right and the j-th row above it by shifting it2j units to the left, j=1,2,… Note that such an operator is actually not a Toeplitz operator. An explicit study of the adjoints and spectra of such operators was carried out in subsequent papers by the same author and there appeared to be an earlier study of related questions done by Latushkin in1980's. Later Arora and Batra considered the generalized slant Toeplitz operators of order k. In fact, k=1yields the standard Toeplitz operators on L2and k=2yields the slant Toeplitz operators considered by Mark. In subsequent papers the authors did much explicit study of this type operators and got the analogous results with slant Toeplitz operators. In Chapter3.1, we discuss the spectrum of generalized slant Toeplitz operator of order k with a polynomial symbol of " length k " and the(U+K)-orbit of it. The main results are the following theorems:Theorem3For fixed k∈Z+, where N≥0, N∈Z and {c-t; N≤t≤N+k-1, t∈Z} be complex numbers and c-N≠0, then the spectrum of(k)Bφ is the closed disc‖φ‖2D and the interior of the disc consists of eigenvalues with infinite multiplicity.Theorem4For fixed k∈Z+, where N∈Z+and{ct; N k+1≤t≤N, t∈Z} be complex numbers and cN≠0, then the spectrum of (K)Bφ is the closed disc‖φ‖2D and the interior of the disc consists of eigenvalues with infinite multiplicity.Theorem5For fixed k G Z+, where N≥0, N∈Z and {c-t; N≤t≤N+k-1, t∈Z} be complex numbers and c-N≠0, then (U+K)((k)Bφ*)={‖φ‖2(V+K);σ{V+K)=D}, where K is a compact operator and V is a pure isometry of infinite multiplicity.Theorem6For fixed k∈Z+, where N∈Z+and {ct; N k+1≤t≤N, t∈Z} be complex numbers and cN≠0, where T is a pure isometry of infinite multiplicity.The generalized slant Toeplitz operator of order k with co-analytic symbol is a upper triangular operator with respect to the basis {zn; n≥0, n∈Z} of H2(T) and its adjoint is in AlgN. We will show that the finite linear combinations of generalized slant Toeplitz operator of order k with co-analytic symbol are closed under addition and multiplication for k∈Z+. So the closure of them is a subalgebra of the upper triangular operators. We get a subalgebra of AlgN which contains all the analytic Toeplitz operators.Let kS denote the set {(k)Bφ;φ∈H∞(T)} and (k)Bφ be the generalized slant Toeplitz operator of order k with symbol if. The main results in Chapter3.2are the following theorems:Theorem7For fixed k, h∈Z+, let (k)Bφ be in kS and (h)Bψ be in hS, then we have (k)Bφ (h)Bψ∈khS.Theorem8Fixed s G Z+, let k1, k2,…,ks be different integers. For1≤t≤s, take Tt∈kt S, then∑t=1s Tt=0if and only if Tt=0,1≤t≤s.Theorem9Let (?)=∑t=1sTt, Tt∈kt S,k1, k2,…, ks be different positive integers, Vs G Z+} and Q denote the norm closure of (?), then (?)is a subalgebra of AlgN.