This chapter is about the most vexing problem in the theory of linear operators on Hilbert space: Invariant Subspace Prob. Subspace invariant The invariant subspace problem. Does every operator on Hilbert space have a nontrivial invariant subspace? Here “operator” means “continuous linear transformation,” and “invariant subspace” means “closed (linear) subspace that the operator takes into itself.” To say that a subspace is “nontrivial” means that it is neither the zero subspace nor the whole space. Space Banach Space normed linear Theorem Schauder Fixed-Point Space Hilbert Examples constructed toward the end of the last century show that in the generality of Banach spaces there do exist operators with only trivial invariant subspaces. For Hilbert space, however, the Invariant Subspace Problem remains open, and is the subject of much research. In this chapter we’ll see why invariant subspaces are of interest and then will prove one of the subject’s landmark theorems: Victor Lomonosov’s 1973 result, a special case of which states: Space Banach Lomonosov, Victor If an operator T on a Banach space commutes with a non-zero compact operator, then T has a nontrivial invariant subspace. This result, which far surpassed anything that seemed attainable at the time, is only part of what Lomonosov proved in an astonishing two-page paper [71] that introduced nonlinear methods—in particular the Schauder Fixed-Point Theorem—into this supposedly hard-core-linear area of mathematics.

An algebra of bounded linear operators on a Hilbert space is called strongly compact whenever each of its bounded subsets is relatively compact in the strong operator topology. The concept is most commonly studied for two algebras associated with a single operator T: the algebra alg(T) generated by the operator, and the operator's commutant com(T). This paper focuses on the strong compactness of these two algebras when T is a composition operator induced on the Hardy space H-2 by a linear fractional self-map of the unit disc. In this setting, strong compactness is completely characterized for alg(T), and "almost" characterized for com(T), thus extending an investigation begun by Fernandez-Valles and Lacruz [A spectral condition for strong compactness, J. Adv. Res. Pure Math. 3 (4) 2011, 50-60]. Along the way it becomes necessary to consider strong compactness for algebras associated with multipliers, adjoint composition operators, and even the Cesaro operator.

The Invariant Subspace Problem (“ISP”) for Hilbert space operators is known to be equivalent to a question that, on its surface, seems surprisingly concrete: For composition operators induced on the Hardy space H 2 by hyperbolic automorphisms of the unit disc, is every nontrivial minimal invariant subspace one dimensional (i.e., spanned by an eigenvector)? In the hope of reviving interest in the contribution this remarkable result might offer to the studies of both composition operators and the ISP, I revisit some known results, weaken their hypotheses and simplify their proofs. Sample results: If φ is a hyperbolic disc automorphism with fixed points at a and β (both necessarily on the unit circle), and C φ the composition operator it induces on H 2, then for every \(f \in \sqrt {(z - \alpha )(z - \beta )}\) H 2, the doubly C φ-cyclic subspace generated by f contains many independent eigenvectors; more precisely, the point spectrum of C φ’s restriction to that subspace intersects the unit circle in a set of positive measure. Moreover, this restriction of C φ is hypercyclic (some forward orbit is dense). Under the stronger restriction \(f \in \sqrt {(z - \alpha )(z - \beta )}\) H p for some p > 2, the point spectrum of the restricted operator contains an open annulus centered at the origin.

The study of composition operators, most notably on the Hardy space H-2, often leads to issues involving Toeplitz operators acting on that space. Toeplitz operators arise, for example, in Littlewood's original proof that composition operators are bounded on H-2, in the computation of their adjoints, and in questions about their "normality." In addition, there are interesting questions about just how "Toeplitz" a composition operator can be. This article reviews some recent work on these matters.

The Invariant Subspace Problem ("ISP") for Hilbert space operators is known to be equivalent to a question that, on its surface, seems surprisingly concrete: For composition operators induced on the Hardy space H^2 by hyperbolic automorphisms of the unit disc, is every nontrivial minimal invariant subspace one dimensional (i.e., spanned by an eigenvector)? In the hope of reviving interest in the contribution this remarkable result might offer to the studies of both composition operators and the ISP, I revisit some known results, weaken their hypotheses and simplify their proofs. Sample results: If f is a hyperbolic disc automorphism with fixed points at a and b (both necessarily on the unit circle), and C_f the composition operator it induces on H^2, then for every function g in the subspace [{(z-a)(z-a)]^(1/2)H^2, the doubly C_f-cyclic subspace generated by g contains many independent eigenvectors; more precisely, the point spectrum of C_f's restriction to that subspace intersects the unit circle in a set of positive measure. Moreover, this restriction of C_f is hypercyclic (some forward orbit is dense).

We study the intertwining relations between analytic Toeplitz operators induced on the Hardy space H^2 by analytic functions bounded on the open unit disc. Our work centers on the connection between intertwining between the Toeplitz operators the image containment between their symbols, as well as on the nature of the intertwining operator. We use our results to study the "extended eigenvalues" of analytic Toeplitz operators, i.e., the special case where the operator is intertwined with a scalar multiple of itself.

We give an elementary proof of a formula recently obtained by Hammond, Moorhouse, and Robbins for the adjoint of a rationally induced composition operator on the Hardy space H2 [Christopher Hammond, Jennifer Moorhouse, Marian E. Robbins, Adjoints of composition operators with rational symbol, J. Math. Anal. Appl. 341 (2008) 626–639]. We discuss some variants and implications of this formula, and use it to provide a sufficient condition for a rationally induced composition operator adjoint to be a compact perturbation of a weighted composition operator.

Nazarov and Shapiro recently showed that, while composition operators on the Hardy space H2 can only trivially be Toeplitz, or even “Toeplitz plus compact,” it is an interesting problem to determine which of them can be “asymptotically Toeplitz.” I show here that if “asymptotically” is interpreted in, for example, the Cesàro (C,α) sense (α>0), then every composition operator on H2 becomes asymptotically Toeplitz.

This paper studies autonomous, single-input, single-output linear control systems on finite time intervals. The object of interest is the output operator O, which associates to each input function and initial state vector the corresponding system output. Main result: If the system has relative degree r<∞, then for any “admissible” Banach space U of inputs, O is a bounded operator taking U×Cn onto the “Sobolev space” of complex functions f∈C(r−1)([0,T]) for which the (r−1)-order derivative f(r−1) is absolutely continuous, with f(r)∈U. This completes recent results of Jönsson and Martin [Ulf Jönsson, Clyde Martin, Approximation with the output of linear control systems, J. Math. Anal. Appl. 329 (2007) 798–821] who showed that if the system is minimal and U is either L2([0,T]) or C([0,T]), then O:U×Cn→U has dense range.

Composition operators on H 2 cannot–except trivially–be Toeplitz, or even ‘Toeplitz plus compact’. However there are natural ways in which they can be ‘asymptotically Toeplitz’. We show here that the study of such phenomena leads to surprising results and interesting open problems. †Dedicated to Professor Peter L. Duren on the occasion of his 70th birthday.

For holomorphic selfmaps of the open unit disc U that are not elliptic automorphisms, the Schwarz Lemma and the Denjoy-Wolff Theorem combine to yield a remarkable result: each such map W has a (necessarily unique) "Denjoy-Wolff point" omega in the closed unit disc that attracts every orbit in the sense that the iterate sequence (phi([n])) converges to w uniformly on compact subsets of U. In this paper we prove that, except for the obvious counterexamples-inner functions having omega is an element of U-the iterate sequence exhibits an even stronger affinity for the Denjoy-Wolff point; phi([n]) -> to in the norm of the Hardy space HP for 1 <= p < infinity. For each such map, some subsequence of iterates converges to w almost everywhere on partial derivative U, and this leads us to investigate the question of almost-everywhere convergence of the entire iterate sequence. Here our work makes natural connections with two important aspects of the study of holomorphic selfmaps of the unit disc: linear- fractional models and ergodic properties of inner functions.

Let T be a continuous linear operator on a Hausdorff topological vector space X over the field C. We show that if T is N-supercyclic, i.e., if X has an N-dimensional subspace whose orbit under T is dense in X, then T* has at most N eigenvalues (counting geometric multiplicity). We then show that N-supercyclicity cannot occur nontrivially in the finite-dimensional setting: the orbit of an N-dimensional subspace cannot be dense in an (N + 1)-dimensional space. Finally, we show that a subnormal operator on an infinite-dimensional Hilbert space can never be N-supercyclic.

These notes supplement the discussion of linear fractional mappings presented in a beginning graduate course in complex analysis. The goal is to prove that a mapping of the Riemann sphere to itself is a rotation if and only if the corresponding map induced on the plane by stereographic projection is a linear fractional whose (two-by-two) coefficient matrix is unitary. 1. Spheres, points, and subspaces 1.1. Point at infinity. Recall that we have discussed two ways of “legitimizing” the “point at infinity” for the complex plane: (a) The Riemann Sphere S2 (cf. our textbook [S, pp. 8–11]). Here the idea is to map the extended plane Ĉ onto the Riemann Sphere S2 via the stereographic projection , making ∞ correspond to the north pole. Recall also that the stereographic projection S\{North Pole} → C is conformal. (b) Complex Projective space CP1 (cf. [S, p. 25]). We regard this as the collection of one dimensional subspaces of C2, with the point z ∈ C identified with the subspace spanned by the column vector [z, 1]t (where the superscript “t” denotes “transpose”), and ∞ identified with the one spanned by [1, 0]t. 1.2. Notation. Let ẑ denote the one dimensional subspace of C2 spanned by the vector [z, 1]t if z ∈ C, and let ∞̂ be the subspace spanned by [1, 0]t. 1.3. Matrices and LFT’s. We have also made a connection between linear fractional transformations and matrices. This begins in a purely formal way by associating each LFT φ(z) = az+b cz+d with the two-by-two complex nonsingular matrix [φ] = [ a b c d ] , noting that actually [φ] should not just stand for one matrix, but for the one-parameter family: all the multiples Date: March 19, 2004.

We characterize the essentially normal composition operators induced on the Hardy space H2 by linear-fractional maps; they are either compact, normal, or (the nontrivial case) induced by parabolic nonautomorphisms. These parabolic maps induce the first known examples of nontrivially essentially normal composition operators. In addition, we characterize those linear-fractionally induced composition operators on H2 that are essentially self-adjoint, and present a number of results for composition operators induced by maps that are not linear-fractional.

We consider, for G a simply connected domain and 0<p<infinity, the Hardy space, H-P(G) formed by fixing a Riemann map tau of the unit disc onto G, and demanding of functions F holomorphic on G that the integrals of \F\(P) over the curves tau({\z\ = r}) be bounded for 0<r<1. The resulting space is usually not the one obtained from the classical Hardy space of the unit disc by conformal mapping. This is reflected in our Main Theorem: H-P (G) supports compact composition operators if and only if partial derivativeG has finite one-dimensional Hausdorff measure. Our work is inspired by an earlier result of Matache (Proc. Amer. Math. Soc. 127 (1999) 1483), who showed that the H-P spaces of half-planes support no compact composition operators. Our methods provide a lower bound for the essential spectral radius which shows that the same result holds with "compact" replaced by "Riesz." We prove similar results for Bergman spaces, with the Hardy-space condition "partial derivativeG has finite Hausdorff 1-measure" replaced by "G has finite area." Finally, we characterize those domains G for which every composition operator on either the Hardy or the Bergman spaces is bounded. (C) 2003 Elsevier Inc. All rights reserved.

We work on the Hardy spaceH2 of the open unit disc\(\mathbb{U}\) and consider the numerical ranges of composition operatorsCφ induced by holomorphic self-maps φ of\(\mathbb{U}\). For maps φ that fix a point of\(\mathbb{U}\) we determine precisely when 0 belongs to the numerical rangeW ofCφ, and in the process discover the following dichotomy: either 0∈W or the real part ofCφ admits a decomposition that reveals it to bestrictly positive-definite. In this latter case we characterize those operators that aresectorial. For compact composition operators our work has the following consequences: it yields a complete description of the corner points of the closure ofW, and it establishes whenW is closed. In the course of our investigation we uncover surprising connections between composition operators, Chebyshev polynomials, and Pascal matrices.

This paper establishes decomposability for composition operators induced on the Hardy spaces H-p (1 less than or equal to p less than or equal to infinity) by parabolic linear fractional self-maps of the unit disc that are not automorphisnis. This result, along with a recent theorem of Miller and Miller [15], shows that no such composition operator is supercyclic. The work here completes part of a previous investigation [1] where the author and Paul Bourdon showed that among linear fractional maps of the disc with no interior fixed point, only the parabolic non-automorphisms induce non-hypercyclic composition operators. Additionally it complements results of Robert Smith [19], who proved decomposability in the case of parabolic automorphisms, and it extends recent work of Gallardo and Montes [6] who used different methods to establish the desired non-supercyclicity for the case p = 2.MCS 2000 Primary 47B33; Secondary 30D55, 47A11, 47A16.

These are notes for a series of lectures that illustrate how continuous linear transformations of infinite dimensional topological vector spaces can have interesting dynamical properties, the study of which forges new links between the theories of dynamical systems, linear operators, and analytic functions.

We investigate the shape of the numerical range for composition operators induced on the Hardy space H2 by conformal automorphisms of the unit disc. We show that usually, but not always, such operators have numerical ranges whose closures are discs centered at the origin. Surprising open questions arise from our investigation.

The backward shift B B on the Bergman space of the unit disc is known to be hypercyclic (meaning: it has a dense orbit). Here we ask: “Which operators that commute with B B inherit its hypercyclicity?” We show that the problem reduces to the study of operators of the form ϕ ( B ) \phi (B) where ϕ \phi is a holomorphic self-map of the unit disc that multiplies the Dirichlet space into itself, and that the question of hypercyclicity for such an operator depends on how freely ϕ ( z ) \phi (z) is allowed to approach the unit circle as | z | → 1 − |z|\to 1- .