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A Hilbert space operator is called universal (in the sense of Rota) if every operator on the Hilbert space is similar to a multiple of the restriction of the universal operator to one of its invariant subspaces. We exhibit an analytic Toeplitz operator whose adjoint is universal in the sense of Rota and which commutes with a quasinilpotent, injective, compact operator with dense range, but unlike other examples, it acts on the Bergman space instead of the Hardy space and this operator is associated with a “hyperbolic” composition operator.

Let X \mathcal {X} be the unital ∗ * -algebra generated by the unilateral shift operator. It is shown that for any nonnegative operator X ∈ X X\in \mathcal {X} there is an element Y ∈ X Y\in \mathcal {X} such that X = Y ∗ Y X=Y^*Y .

We study the backward shift operator on Hilbert spaces H α {\mathcal {H}}_{\alpha } (for α ≥ 0 {\alpha \geq 0} ) which are norm equivalent to the Dirichlet-type spaces D α D_{\alpha } . Although these operators are unitarily equivalent to the adjoints of the forward shift operator on certain weighted Bergman spaces, our approach is direct and completely independent of the standard Cauchy duality. We employ only the classical Hardy space theory and an elementary formula expressing the inner product on H α {\mathcal {H}}_{\alpha } in terms of a weighted superposition of backward shifts.

In the following, we consider some cases where the point spectrum of an operator is either very stable or very unstable with respect to small perturbations of the operator. The main result is about the shift operator on l 2 , l_2, whose point spectrum is what we will call strongly stable. We also give some general perturbation results, including a result about the size of the set of operators that have an eigenvalue.

This note examines the subspaces of the space of Cauchy transforms of measures on the unit circle that are invariant under the backward shift operator f → z − 1 ( f − f ( 0 ) ) f \to z^{-1}(f - f(0)) . We examine this question when the space of Cauchy transforms is endowed with both the norm and weak ∗ {}^* topologies.

All wavelets can be associated to a multiresolution-like structure, i.e. an increasing sequence of subspaces of L 2 (R). We consider the interaction of a wavelet and the shift operator in terms of which of the subspaces in this multiresolution-like structure are invariant under the shift operator. This action defines the notion of the shift invariance property of order n. In this paper we show that wavelets of all levels of shift invariance exist, first for the classic case of dilation by 2, and then for arbitrary integral dilation factors.

We study the decay at large distances of operator kernels of functions of generalized Schrodinger operators, a class of semibounded second order partial differential operators of mathematical physics, which includes the Schrodinger operator, the magnetic Schrodinger operator, and the classical wave operators (i.e., acoustic operator, Maxwell operator, and other second order partial differential operators associated with classical wave equations). We derive an improved Combes-Thomas estimate, obtaining an explicit lower bound on the rate of exponential decay of the operator kernel of the resolvent. We prove that for slowly decreasing smooth functions the operator kernels decay faster than ally polynomial.

It is well known that for every isometry V, tr[V*, V] = -ind(V). This fact for the shift operator is a basis for many important developments in operator theory and topology. In this paper we prove an analogous formula for a pair of isometries (V 1 , V 2 ), namely tr[V* 1 , V 1 , V* 2 , V 2 ) = -2ind(V 1 ,V 2 ), where [V* 1 , V 1 , V* 2 , V 2 ] is the complete anti-symmetric sum and ind(V 1 , V 2 ) is the Fredholm index of the pair (V 1 , V 2 ). The major tool is what we call the fringe operator. Two examples are considered.

For the general one dimensional Schrodinger operator - d 2 /dx 2 +q (x) with real q we study some analytic aspects related to order-one trace formulas originally due to Buslaev-Faddeev, Faddeev-Zakharov, and Gesztesy-Holden-Simon-Zhao. We show that the condition q E L 1 (R) guarantees the existence of the trace formulas of order one only with certain resolvent regularizations of the integrals involved. Our principle results are simple necessary and sufficient conditions on absolute summability of the formulas under consideration. These conditions are expressed in terms of Fourier transforms related to q.

A complete description of the reducing subspaces of weighted unilateral shift operators of finite multiplicity is obtained.

We consider a matrix differential operator with singular entries which arises in magnetohydrodynamics. By means of the asymptotic Hain-Lüst operator and some pseudo-differential operator techniques, we determine the essential spectrum of this operator. Whereas in the regular case, the essential spectrum consists of two intervals, it turns out that in the singular case two additional intervals due to the singularity may arise. In addition, we establish criteria for the essential spectrum to lie in the left half-plane.

Let T T be a bounded operator on a Hilbert space H , \mathfrak {H}, and A , B A,B positive definite operators. Kato has shown that if ‖ T x ‖ ≤ ‖ A x ‖ \Vert T x \Vert \leq \Vert A x \Vert and ‖ T ∗ y ‖ ≤ ‖ B y ‖ \Vert T^{*} y \Vert \leq \Vert B y \Vert for all x , y ∈ H x, y \in \mathfrak {H} , then | ( T x , y ) | ≤ | | f ( A ) x | | | | g ( B ) y | | , |(Tx , y)| \leq ||f(A)x|| \; ||g(B)y||, where f ( t ) , g ( t ) f(t), g(t) are operator monotone functions defined on [ 0 , ∞ ) [0, \infty ) such that f ( t ) g ( t ) = t f(t) g(t)=t . Furuta has shown that | ( T | T | α + β − 1 x , y ) | ≤ | | A α x | | | | B β y | | , where 0 ≤ α , β ≤ 1 , 1 ≤ α + β . |(T|T|^{\alpha +\beta -1}x,y)|\leq ||A^{\alpha }x|| ||B^{\beta }y||, \text { where } 0 \leq \alpha ,\beta \leq 1, 1\leq \alpha + \beta . Let f ( t ) , g ( t ) f(t), g(t) be any continuous operator monotone functions, and set h ( t ) = f ( t ) g ( t ) / t h(t) = f(t) g(t) /t for t > 0. t >0. We will show that T h ( | T | ) Th(|T|) is well defined and | ( T h ( | T | ) x , y ) | ≤ | | f ( A ) x | | | | g ( B ) y | | . |(T h(|T|)x,y)| \leq ||f(A)x|| \; ||g(B)y||. Moreover, we will extend this result for unbounded closed operators densely defined on H . \mathfrak {H}.

It is shown that an n n –tuple of bounded linear operators on a complex Hilbert space, which is positive definite in the sense of Halmos, must be commutative. Some generalizations of this result to the case of pairs of unbounded operators are obtained.

A sufficient condition for the stability of the range of a positive operator in a Hilbert space is given. As a consequence, we get a class of additive perturbations which preserve regularity of the critical point 0 of a positive operator in a Krein space.

We give a characterization of the pairs of weights (u, v) such that the Weyl fractional integral operator maps Lp(vdx) into weak Lq(udx), 1 < p ≤ q < ∞ or p = 1 < q < ∞. For the case p < q we give necessary and sufficient conditions for the weak type of a maximal operator that includes as particular cases the Weyl fractional integral, the dual of the Hardy operator and the fractional one-sided maximal operator. As a consequence we give a new characterization of the pairs of weights for which the fractional one-sided maximal operator is bounded.

We introduce and discuss a class of operators, to be referred to as the class of completely hyperexpansive operators, which is in some sense antithetical to the class of contractive subnormals. The new class is intimately related to the theory of negative definite functions on abelian semigroups. The known interplay between positive and negative definite functions from the theory of harmonic analysis on semigroups can be exploited to reveal some interesting connections between subnormals and completely hyperexpansive operators. If HE is a complex infinite-dimensional separable Hilbert space, then IB(IH) will denote the set of bounded linear operators on H1. Recall that S in IB(IH) is said to be subnormal if there exist a Hilbert space IK containing JHE and a normal operator N in IB(K) such that NHI C Hl and N/I = S. For all of the elementary results pertaining to subnormals (and related classes of operators such as hyponormals) that are stated below without proof, the reader is referred to [Co]. If {en}n>O is an orthonormal basis for HI, then a weighted shift operator T on Hl with the weight sequence {c}a n>o is defined through the relations Ten = anen+1 (n > 0). We will always assume that an > 0 for all n. An excellent reference for the basic properties of weighted shifts is [S]. We will often use the notation T {an} to indicate a weighted shift. Of particular interest to us are the shifts U {1} (the unilateral shift), B: { n?1/ n-2} (the Bergman shift) and D : {f /n 2/ /n+1} (the Dirichlet shift). The shifts U and B are subnormal with U in fact being an isometry; while D is a 2-isometry (I 2D*D + D*2D2 = 0, I and 0 being the identity and zero operator respectively). It is the purpose of this note to introduce and discuss a class of operators that is in some sense antithetical to the class of contractive subnormals. The connections between the theory of positive definite functions on abelian semigroups and the theory of subnormals are well-known and will be touched upon briefly in the sequel. The new class of operators, to be referred to as the class of completely hyperexpansive operators, is intimately related to the theory of negative definite functions on abelian semigroups. The shift U and the shift D are rather special examples of the new class. The known connections between positive and negative definite functions will allow us to correlate U and D as well as B and D in meaningful ways. Using the idea of the Laplace Transform, we will be able to associate with every hyperexReceived by the editors May 17, 1995. 1991 Mathematics Subject Classification. Primary 47B20; Secondary 47B39.

We give a very elementary proof of the reverse Hölder type inequality for the classes of weights which characterize the boundedness on L p L^{p} of the Hardy operator for nonincreasing functions. The same technique is applied to Calderón operator involved in the theory of interpolation for general Lorentz spaces. This allows us to obtain further consequences for intermediate interpolation spaces.

This note studies more deeply the results obtained in an earlier paper of the author (An operator-valued moment problem, Proc. Amer. Math. Soc. 112 (1991)). It gives a similar condition for the solvability of the L-problem of moments, using the operator phase shift. Based on this, it underlines some of the aspects of the operator phase shift used in the L-problem of moments.

We give an abstract matrix norm characterization for operator algebras with contractive approximate identities by using the second dual approach. We show that if A is an L ∞ {L^\infty } -Banach pseudoalgebra with a contractive approximate identity, then the second dual A ∗ ∗ {A^{ \ast \ast }} of A is a unital L ∞ {L^\infty } -Banach pseudoalgebra containing A as a subalgebra. It follows from the Blecher-Ruan-Sinclair characterization theorem for unital operator algebras that A ∗ ∗ {A^{ \ast \ast }} is completely isometrically unital isomorphic to a concrete unital operator algebra on a Hilbert space. Thus A is completely isometrically isomorphic to a concrete nondegenerate operator algebra with a contractive approximate identity.

In this paper we consider the integration operator in two variables on L 2 [ 0 , 1 ] 2 {L_2}{[0,1]^2} , determine its multiplicity and reducing subspaces, and make some observations about its invariant and hyperinvariant subspaces.

A direct proof is given for the matricial norm characterization of operator spaces.

Suppose Φ \Phi is an appropriate Young’s function and w ( x ) , v ( x ) w(x),v(x) are nonnegative locally integrable functions. Let T T denote one of three linear operators of special importance that map suitable functions on R n {R^n} into functions on R n {R^n} . For the Hardy operator T T , we study the inequality \[ ∫ 0 ∞ Φ ( | T f ( x ) | ) w ( x ) d x ⩽ C ∫ 0 ∞ Φ ( | f ( x ) | ) v ( x ) d x \int _0^\infty {\Phi (|Tf(x)|)w(x)\,dx \leqslant C\int _0^\infty {\Phi (|f(x)|)v(x)\,dx} } \] and for the Hardy-Littlewood maximal operator or fractional integrals T T , we discuss the inequalities \[ ∫ R n Φ ( | T ( f v ) ( x ) | ) w ( x ) d x ⩽ C ∫ R n Φ ( | f ( x ) | ) v ( x ) d x . \int _{{R^n}} {\Phi (|T(fv)(x)|)w(x)\,dx \leqslant C\int _{{R^n}} {\Phi (|f(x)|)v(x)\,dx.} } \] In all cases we obtain the necessary and sufficient conditions.

The composition operator induced by a hyperbolic Möbius transform ϕ \phi on the classical Hardy space H 2 {H^2} is considered. It is known that the invariant subspace problem for Hilbert space operators is equivalent to the fact that all the minimal invariant subspaces of this operator are one- dimensional. In connection with that we try to decide by the properties of a given function u u in H 2 {H^2} if the corresponding cyclic subspace is minimal or not. The main result is the following. If the radial limit of u u is continuously extendable at one of the fixed points of ϕ \phi and its value at the point is nonzero, then the cyclic subspace generated by u u is minimal if and only if u u is constant.

In this paper we derive a first-order differential operator that can serve as an alternative to the non-Euclidean wave operator to study S1( 2 , R ) {\text {S1(}}2,R) .

This note provides a complete description of the spectral picture of a hyponormal bilateral operator weighted shift of finite multiplicity. If the weights are m × m m \times m matrices, then the operator cannot by cyclic, unless it is a normal operator of a very special kind. An example shows that, nevertheless, there exists a cyclic, nonnormal, hyponormal bilateral operator weighted shift whose weights are operators acting on an infinite dimensional Hilbert space. (This answers a question of J. B. Conway.)

Let 0 ≤ T : L p ( Y , ν ) → L q ( X , μ ) 0 \leq T:{L^p}(Y,\nu ) \to {L^q}(X,\mu ) be a positive linear operator and let | | T | | p , q ||T|{|_{p,q}} denote its operator norm. In this paper a method is given to compute | | T | | p , q ||T|{|_{p,q}} exactly or to bound | | T | | p , q ||T|{|_{p,q}} from above. As an application the exact norm | | V | | p , q ||V|{|_{p,q}} of the Volterra operator V f ( x ) = ∫ 0 x f ( t ) d t Vf(x) = \int _0^x {f(t)dt} is computed.

Let S ∗ {S^*} denote the backward shift operator on the Hardy space H 2 {H^2} of the unit disk. A subspace M M of H 2 {H^2} is called nearly invariant if S ∗ h {S^*}h is in M M whenever h h belongs to M M and h ( 0 ) = 0 h(0) = 0 . In particular, the kernel of every Toeplitz operator is nearly invariant. A function theoretic characterization is given of those nearly invariant subspaces which are the kernels of Toeplitz operators, and it is shown that they can be put into one-to-one correspondence with the Cartesian product of the set of exposed points of the unit ball of H 1 {H^1} with the set of inner functions.

Necessary and sufficient for the closure of a linear subspace to lie in the range of a bounded linear operator is a certain "bounded preimage property" for the operator. If T: X -Y is a bounded linear operator between normed spaces then we shall, par abus de notation, also write [3] (0.1) T: loo(X) * loo(Y) for the operator induced between the corresponding spaces of bounded vectorvalued sequences (0.2) lw (X) = {x E XN: sup l lxn ll 0 for which dist(x, T-(O)) < kITxI for eachx ET-1(M), so that if y E lo (M) is arbitrary then there is x E XN for which y = Tx with dist(xn, T-'(0)) < kilYn I, and then z E T-l(0)N for which llx-zll < 2 dist(x, T-I(0)), giving (1.3) y=T(x-z) with x-z E loo(X). Received by the editors September 1, 1991. 1991 Mathematics Subject Classification. Primary 47A05; Secondary 47B07, 46B08.

It has been conjectured that a certain operator T T belonging to the group G \mathcal {G} of invertible elements of the algebra Alg ⁡ Z \operatorname {Alg} {\mathbf {Z}} of doubly infinite upper-triangular bounded matrices lies outside the connected component of the identity in G \mathcal {G} . In this note we show that T T actually lies inside the connected component of the identity of G \mathcal {G} .

Let be a real continuous function on an interval, and consider the operator function defined for Hermitian operators . We will show that if is increasing w.r.t. the operator order, then for the operator function is convex. Let and be functions defined on an interval . Suppose is non-decreasing and is increasing. Then we will define the continuous kernel function by , which is a generalization of the Lowner kernel function. We will see that it is positive definite if and only if whenever for Hermitian operators , and we give a method to construct a large number of infinitely divisible kernel functions.

A short proof of the result that if P P is a positive compact operator and A A is a contraction such that p ≤ A ∗ P A p \leq {A^*}PA , then P = A ∗ P A , ran ¯ P P = {A^*}PA,\overline {\operatorname {ran} } P reduces A A and A | ran ¯ P A|\overline {\operatorname {ran} } P is unitary is given.

Sufficient conditions on a sequence are given in order that the linear manifold spanned by its right translates is dense in certain Hilbert spaces of sequences.

Consider the operator equation ( ∗ ) A X + X B = Q (\ast )AX + XB = Q ; here A and B are (possibly unbounded) selfadjoint operators and Q is a bounded operator on a Hilbert space. The theory of one parameter semigroups of operators is applied to give a quick derivation of M. Rosenblum’s formula for approximate solutions of ( ∗ ) (\ast ) . Sufficient conditions are given in order that ( ∗ ) (\ast ) has a solution in the Schatten-von Neumann class C p {\mathcal {C}_p} if Q is in C p {\mathcal {C}_p} . Finally a sufficient condition for solvability of ( ∗ ) (\ast ) is given in terms of T. Kato’s notion of smoothness.

Operator radii w ρ ( T ) {w_\rho }(T) for a bounded linear operator T T on a Hilbert space were introduced in connection with unitary ρ \rho -dilations. We shall be concerned with universal estimates for the ratios \[ w ρ ( S T ) / ( w σ ( S ) ⋅ w ρ ( T ) ) {w_\rho }(ST)/({w_\sigma }(S) \cdot {w_\rho }(T)) \] for commuting operators S , T S,T and σ , ρ > 0 \sigma ,\rho > 0 .

Let A be a fixed normal operator and let N ( Λ ) \mathfrak {N}(\Lambda ) denote the normal operators with spectrum contained in Λ \Lambda . Provided there is some N in N ( Λ ) \mathfrak {N}(\Lambda ) such that A − N A - N belongs to the Schatten class c p , p ⩾ 2 {c_p},p \geqslant 2 , the main result of this paper obtains a best approximation for A from N ( Λ ) \mathfrak {N}(\Lambda ) with respect to the Schatten p-norm. A necessary and sufficient condition is given for A to have a unique best approximation in that case.

T. B. Hoover has shown that if A is a reductive operator, then A = A 1 ⊕ A 2 A = {A_1} \oplus {A_2} , where A 1 {A_1} is normal and all the invariant subspaces of A 2 {A_2} are hyperinvariant. A new proof is presented of this result, and several corollaries are derived. Among these is the fact that if A is hyperinvariant and T is polynomially compact and A T = T A AT = TA , then A ∗ T = T A ∗ {A^ \ast }T = T{A^ \ast } . It is also shown that every reductive operator is quasitriangular.

Let XC be a separable, infinite dimensional, complex Hilbert space, and let E(9C) denote the algebra of all bounded linear operators on SC. In [2] it was shown that every quasinilpotent operator T in E(jC) (i.e., every T whose spectrum is the singleton {O}) is a norm-limit of nilpotent operators. The purpose of this note is to give a very short and easily remembered proof of this theorem. Our proof has two basic ingredients: the use of a model for quasinilpotent operators constructed in [4], and the use of a deep theorem of Voiculescu [7]. For completeness we review the relevant facts from [4] and [7] that we shall need. Let X. = X ED X e ... be the direct sum of S0 copies of SC indexed by the positive integers, let K = K,,' I= be a monotone decreasing sequence of nonnegative numbers converging to zero, and let UK denote the quasinilpotent backward weighted shift operator in Ef(f3C) defined by the equation UK(X19 X2, * * * 9 Xn9 .. . ) = (K1X2, K2X3,. . .9 KnXn, 19 . .. ). (1) THEOREM A [4]. If T is any quasinilpotent operator in e(9C), then there is a sequence K = K(T) = {Kn,) 'I with the above described properties and a subspace 91 of 'K.C that is invariant under the quasinilpotent operator UK such that T is similar to UK191 and such that the subspace 'CK e )1 is infinite dimensional.

The spherical maximal operator for radial functions of R n {{\mathbf {R}}^n} is shown to be a restricted weak type L p {L^p} bounded operator for p = n / ( n − 1 ) p = n/(n - 1) . The proof uses methods for restricted weak type single weight norm inequalities.

It is shown that a monotone operator function f defined on an open subset A of the real numbers may be extended to a monotone operator function on the convex hull of A. Let f be a bounded real-valued Borel-measurable function defined on a Borel subset A of the real numbers. Let A be a bounded selfadjoint operator on a separable complex Hilbert space H such that the spectrum a(A) of A is contained in A. Then A(A) is the selfadjoint operator on H defined by ff(X)K , 0, p e H, where E is the resolution of the identity corresponding to A. The functionfis said to be a monotone operator function on A if f(A) < f(B) whenever A and B are bounded selfadjoint operators on a Hilbert space H such that a(A) C A, a(B) C A, and A < B. If f satisfies this monotonicity condition for the totality of finite-dimensional complex Hilbert spaces, then f is said to be a monotone matrix function on A. The following characterization of monotone operator functions is essentially due to Loewner [5]. Loewner proved the theorem for monotone matrix functions; Bendat and Sherman [1] proved that a function f is monotone matrix on an open interval (a,b) if and only if f is monotone operator on (a,b). THEOREM 1. A real-valued function f defined on an open interval (a,b) is a monotone operator function on (a,b) if and only if f is analytic on (a,b), can be analytically continued onto the upper half-plane, and represents there a holomorphic function with nonnegative imaginary part. It will be shown that there is a similar characterization of monotone operator functions defined on an arbitrary open subset A of the real numbers. Let f be a monotone operator function on an open subset A of the real numbers. By Theorem 1, f is continuously differentiable on A; see also [4, p. 73]. Associated with f is the kernel K, defined on A x A by Loewne madeeK(xtei fv ue f() (t k Ar)e K(i,s ) = f mt(). Loewner made extensive use of this kernel in his study of monotone matrix Received by the editors May 21, 1975. AMS (MOS) subject classifications (1970). Primary 47A60, 47B 15; Secondary 30A14, 30A76.

Shift analysis includes abstract treatments of inner-outer factorization problems, the factorization problem for nonnegative functions on a circle, and Szegö’s infimum problem (for scalar or operator valued functions). These problems are here generalized to a setting where the shift operator is replaced by a pair of isometries.

Kato has proven L 2 {L^2} -boundedness if the symbol a ( x , z ) a(x,z) is such that | D x β D z α a ( x , z ) | ⩽ ( constant ) ( 1 + | z | ) ( | β | − | α | ) ρ |D_x^\beta D_z^\alpha a(x,z)|\; \leqslant ({\text {constant}}){(1 + |z|)^{(|\beta | - |\alpha |)\rho }} for | α | ⩽ [ n / 2 ] + 1 , | β | ⩽ [ n / 2 ] + 2 |\alpha | \leqslant [n/2] + 1,|\beta | \leqslant [n/2] + 2 and 0 > ρ > 1 0 > \rho > 1 . In this paper, L 2 {L^2} -boundedness is shown for a corresponding Hölder continuity condition which requires slightly less smoothness for a ( x , z ) a(x,z) .

A characterization of the operators A A for which the equation T X − X V = A TX - XV = A is solvable is given, where T T is a fixed right invertible operator and V V is a fixed unilateral shift.

We give a new proof of a result of Chandler which shows that a monotone operator function defined on a set J admits an analytic continuation to the upper and lower half-planes, and that this continuation is a Pick function, real and regular on the convex hull of J. Let J be a subset of the real axis and f(x) a real-valued function defined on J. If H is a selfadjoint operator on Hilbert space having a spectrum contained in J, the operator f(H) is defined by the usual operational calculus

Calculation of the numerical range of a weighted shift is reduced to the solution of a polynomial equation when the weights form a periodic sequence, or approach a periodic sequence from below. Introduction. A weighted shift on 12 or 12 is a linear operator S defined by Sen = snen+1 where fen} is an orthonormal basis, and {Sn} a sequence of complex numbers. The numerical range of an operator S is the set of complex numbers (Sx, x) where llxll = 1; this is denoted W(S). For definiteness we assume here a one-sided shift, indexed by positive integers; the proofs and results are the same for a two-sided shift. Then we are given x = x1e, + x2e2 + , xi complex, I Ixi 12 0, and we shall do so. THEOREM 1. If {sn} is a periodic sequence, of period r, then w(S) = maxt s1x1 x2 + S2X2X3 +? * + Srxrxl: xi real, x2 + ... + x 2=1} and finding this is equivalent to solving a polynomial equation of degree r. Received by the editors April 8, 1974 and, in revised form, April 18, 1975. AMS (MOS) subject classailcations (1970). Primary 47A10, 47B99.

An exact sequence resolving the Elder operator of the calculus of variations for polynomial differential equations in one independent and one dependent variable is described. This resolution provides readily verifiable necessary and sufficient conditions for such a polynomial differential equation to be the Euler equation for some Lagrangian. An explicit construction of the Lagrangian is given.

A unilateral weighted shift T T on a Hilbert space H H is an operator such that T e n = w n e n + 1 T{e_n} = {w_n}{e_{n + 1}} for some orthonormal basis { e n } n = 0 ∞ \left \{ {{e_n}} \right \}_{n = 0}^\infty and weight sequence { w n } n = 0 ∞ \left \{ {{w_n}} \right \}_{n = 0}^\infty . If we assume w n > 0 {w_n} > 0 , for all n n , and let β ( n ) = w 0 ⋯ w n − 1 \beta \left ( n \right ) = {w_0} \cdots {w_{n - 1}} for n > 0 n > 0 and β ( 0 ) = 1 \beta \left ( 0 \right ) = 1 , then T T is unitarily equivalent to f ↦ z f f \mapsto zf on the weighted space H 2 ( β ) {H^2}\left ( \beta \right ) of formal power series ∑ n = 0 ∞ f ^ ( n ) z n \sum \nolimits _{n = 0}^\infty {\hat f\left ( n \right ){z^n}} such that ∑ n = 0 ∞ | f ^ ( n ) | 2 [ β ( n ) ] 2 > ∞ \sum \nolimits _{n = 0}^\infty {{{\left | {\hat f\left ( n \right )} \right |}^2}{{\left [ {\beta \left ( n \right )} \right ]}^2} > \infty } . Regarding T T as multiplication for z z on the space H 2 ( β ) {H^2}\left ( \beta \right ) , it is shown that, if w n ↑ 1 {w_n} \uparrow 1 and f f is analytic in a neighborhood of the unit disk, then either f f is cyclic for T ∗ {T^ * } or f f is contained in a finite-dimensional T ∗ {T^ * } -invariant subspace. This was shown—by different methods— for the unweighted shift operator by Douglas, Shields, and Shapiro [2]. It is also shown that every finite-dimensional T ∗ {T^ * } -invariant subspace is of the form \[ ( ( z − α 1 ) n 1 ⋯ ( z − α k ) n k H 2 ( β ) ) ⊥ , {\left ( {{{\left ( {z - {\alpha _1}} \right )}^{{n_1}}} \cdots {{\left ( {z - {\alpha _k}} \right )}^{{n_k}}}{H^2}\left ( \beta \right )} \right )^ \bot }, \] for some α 1 , … , α k {\alpha _1}, \ldots ,{\alpha _k} in the unit disk and n 1 , … , n k {n_1}, \ldots ,{n_k} positive integer.

Let H H denote a separable, infinite-dimensional complex Hilbert space. A two-sided ideal I I of operators on H H possesses the generalized Fuglede property (GFP) if, for every normal operator N N and every X ∈ L ( H ) X \in L(H) , N X − X N ∈ I NX - XN \in I implies N ∗ X − X N ∗ ∈ I {N^ * }X - X{N^ * } \in I . Fuglede’s Theorem says that I = { 0 } I = \left \{ 0 \right \} has the GFP. It is known that the class of compact operators and the class of Hilbert-Schmidt operators have the GFP. We prove that the class of finite rank operators and the Schatten p p -classes for 0 > p > 1 0 > p > 1 fail to have the GFP. The operator we use as an example in the case of the Schatten p p -classes is multiplication by z + w z + w on L 2 {L^2} of the torus.