This paper introduces a class of Integro-multiplication operators on Hilbert spaces of analytic functions with reproducing kernels of the form K phi (z,w) = infinity n-ary sumation n=0 f(z)f(w) with f(z) = (n + 1)zn +phi(z)zn+1,where phi is an element of H infinity(D). Hyponormality and subnormality of the operators is explored in some special cases, particularly the case where phi(z) = 1. Additionally the idea of M-dominating matrices is introduced as a means of establishing the norms of these operators.

A convex polynomial is a convex combination of the monomials {1, x, x(2), ... }. This paper establishes that the convex polynomials on R are dense in L-p(mu) and weak* dense in L-infinity(mu) whenever mu is a compactly supported regular Borel measure on R and mu([-1, infinity)) = 0. It is also shown that the convex polynomials are norm dense in C(K) precisely when K boolean AND [-1, infinity) = empty set, where K is a compact subset of the real line. Moreover, the closure of the convex polynomials on [-1,b] is shown to be the functions that have a convex power series representation. A continuous linear operator T on a locally convex space X is convex-cyclic if there is a vector x is an element of X such that the convex hull of the orbit of x is dense in X. The previous results are used to characterize which multiplication operators on various real Banach spaces are convex-cyclic. Also, it is shown for certain multiplication operators that every nonempty closed invariant convex set is a closed invariant subspace.

We define a convex-polynomial to be one that is a convex combination of the mono-mials {1, z, z(2),...}. This paper explores the intimate connection between peaking convex-polynomials, interpolating convex-polynomials, invariant convex sets, and the dynamics of matrices. In particular, we use these intertwined relations to both prove which matrices are convex-cyclic while at the same time proving that we can prescribe the values and a finite number of the derivatives of a convex-polynomial subject to certain natural constraints. These properties are also equivalent to determining those matrices whose nonempty invariant closed convex sets are all invariant subspaces. Our characterization of the convex-cyclic matrices gives a new and correct proof of a similar result by Rezaei that was stated and proven incorrectly.

We give a Hahn–Banach characterization for convex-cyclicity. We also obtain an example of a bounded linear operator S on a Banach space with σp(S⁎)=∅ such that S is convex-cyclic, but S is not weakly hypercyclic and S2 is not convex-cyclic. This solved two questions of Rezaei in [25] when σp(S⁎)=∅. We also characterize the diagonalizable normal operators that are convex-cyclic and give a condition on the eigenvalues of an arbitrary operator for it to be convex-cyclic. We show that certain adjoint multiplication operators are convex-cyclic and show that some are convex-cyclic but no convex polynomial of the operator is hypercyclic. Also some adjoint multiplication operators are convex-cyclic but not 1-weakly hypercyclic.

A convex-polynomial is a convex combination of the monomials $\{1, x, x^2, \ldots\}$. This paper establishes that the convex-polynomials on $\mathbb R$ are dense in $L^p(\mu)$ and weak$^*$ dense in $L^\infty(\mu)$, precisely when $\mu([-1,\infty)) = 0$. It is shown that the convex-polynomials are dense in $C(K)$ precisely when $K \cap [-1, \infty) = \emptyset$, where $K$ is a compact subset of the real line. Moreover, the closure of the convex-polynomials on $[-1,b]$ are shown to be the functions that have a convex-power series representation. A continuous linear operator $T$ on a locally convex space $X$ is convex-cyclic if there is a vector $x \in X$ such that the convex hull of the orbit of $x$ is dense in $X$. The above results characterize which multiplication operators on various real Banach spaces are convex-cyclic. It is shown for certain multiplication operators that every closed invariant convex set is a closed invariant subspace.

We consider analytic reproducing kernel Hilbert spaces H with orthonormal bases of the formf(an + bnz)z n : n 0g. If bn = 0 for all n, then H is a diagonal space and multiplication by z, Mz, is a weighted shift. Our focus is on providing extensive classes of examples for which Mz is a bounded subnormal operator on a tridiagonal spaceH where bn6 0. The Aronszajn sum of H and (1 z)H where H is either the Hardy space or the Bergman space on the disk are two such examples.

If X is a locally convex topological vector space over a scalar field F = R or C and if E is a subset of X, then we define E to be it-weakly dense in X if for every onto continuous linear operator F : X -> F-n we have that F(E) is dense in F-n. If X is a Hilbert space, this is equivalent to requiring that E have a dense orthogonal projection onto every subspace of dimension a. We then consider continuous linear operators on X that have orbits or scaled orbits that are n-weakly dense in X. We show that on a separable Hilbert space there are non-trivial examples of such operators and establish many of their basic properties. A fundamental tool is Ball's solution of the complex plank problem which implies that certain sets arc 1-weakly closed. (C) 2012 Elsevier Inc. All rights reserved.

We define an operator to n-weakly hypercyclic if it has an orbit that has a dense projection onto every n-dimensional subspace. Similarly, an operator is n-weakly supercyclic if it has a scaled orbit that has a dense projection onto every n-dimensional subspace. In this paper, we show the following results: (i) There are no n-weakly hypercyclic matrices on \({\mathbb{R}^n}\) or \({\mathbb{C}^n}\). (ii) There are no 2-weakly supercyclic matrices on \({\mathbb{C}^n}\) for n ≥ 2. (iii) There are no 3-weakly supercyclic matrices on \({\mathbb{R}^n}\) for n ≥ 3; and (iv) there are 2-weakly supercyclic matrices on \({\mathbb{R}^n}\) if and only if n is even. Finally, we show that there is an onto isometry on \({\ell^2_\mathbb{R}(\mathbb{N})}\) that is 2-weakly supercyclic, but not 3-weakly supercyclic and also give some examples involving tuples of matrices. We conclude with some questions.

In this paper we present the highlights of the theory of subnormal operators that was initiated by Paul Halmos in 1950. This culminates in Thomson's Theorem on bounded point evaluations where several applications are presented. Throughout the paper are some open problems.

It turns out that we can also describe the nearly invariant subspaces of in terms of a de Branges-type space on . First let us review the well-known de Branges spaces on ℂ ∖ ℝ . We follow [25, p. 9–12]. Let Ψ be an analytic function on the upper half plane $$ \mathbb{C}_ + = \{ \Im z > 0\} $$ such that ℜΨ≥0. The classical Herglotz theorem [25, p. 7] says that there is a non-negative measure μ on $$ \mathbb{R} $$ and a non-negative number p such that 5.1.1 $$ \Re \Psi (x + iy) = py + \frac{1} {\pi }\int_{ - \infty }^\infty {\frac{y} {{(t - x)^2 + y^2 }}} d\mu (t), x + iy \in \mathbb{C}_ + . $$ The reader will recognize the above integral as the Poisson integral of μ. Extend Ψ to the lower half plane so that $$ \Psi (z) = \overline {\Psi (\bar z),} z = x + iy, y < 0. $$ . A theorem of de Branges [25, p. 9] says that there exists a unique Hilbert space L(Ψ) of analytic functions on $$ \mathbb{C}\backslash \mathbb{R} $$ such that for each fixed $$ w \in \mathbb{C}\backslash \mathbb{R} $$ , the function 5.1.2 $$ z \mapsto \frac{{\Psi (z) + \overline {\Psi (w)} }} {{\pi i(\bar w - z)}} $$ belongs to L(Ψ) and 5.1.3 $$ F(w) = \left\langle {F(z),\frac{{\Psi (z) + \overline {\Psi (w)} }} {{\pi i(\bar w - z)}}} \right\rangle _{\mathcal{L}(\Psi )} \forall F \in \mathcal{L}(\Psi ). $$ The previous identity says that the functions in (5.1.2) are the reproducing kernel functions for L(Ψ). Furthermore, if μ is the measure from (5.1.1), the linear transformation 5.1.4 $$ f \mapsto \frac{1} {{\pi i}}\int_{ - \infty }^\infty {\frac{{f(t)}} {{t - z}}} d\mu (t) $$ maps L2 (μ) isometrically into L(Ψ) and the orthogonal complement of the range of this transformation contains only constant functions. For example, if p=0 in (5.1.1), this map is onto.

In this section we use our main theorem about nearly invariant subspaces (Theorem 3.1.2) and the conformal map from (2.3.8) to give a full description of the invariant subspaces (under Sf=zf) of H2 (G). Let us get started with a few preliminary observations.

If B(H) is the algebra of bounded linear operators on a Hilbert space (H) and K is the ideal of compact operators on (H), one forms the Calkin algebra B(H)/K and the natural map π: B(H) → B(H)/K. Recall that A∈B(H) is Fredholm if π(A) is invertible in B(H)/K. A well-known theorem [19, p. 356] says that A is Fredholm precisely when Rng A is closed and both ker A and H/RangA are finite dimensional. An operator A is semi-Fredholm if π(A) is either right or left invertible in B(H)/K. Equivalently. A is semi-Fredholm if and only if RngA is closed and either ker(A) or H/RngA is finite dimensional. We also use the notation $$ \sigma (A): = \{ \lambda \in \mathbb{C}:\lambda I - A is not invertible\} (spectrum of A),$$ $$ \sigma (A): = \{ \lambda \in \mathbb{C}:\lambda I - A is not Fredholm\} (essential spectrum of A).$$ Note that σ e (A) ⊂ σ(A). For a semi-Fredholm operator A let $$ ind(A): = dim ker A - dim (H/Rng A)$$ be the index of A. When the set ℤ∪{±∞| is endowed with the discrete topology, the map A→ ind(A) (from the set of semi-Fredholm operators to ℤ∪{±∞| is continuous [19, p. 361].

If H is a Hilbert space and T : H ? H is a continous linear operator, a natural question to ask is: What are the closed subspaces M of H for which T M ? M? Of course the famous invariant subspace prob

We now extend our main theorem (Theorem 6.2.1) to domains with several slits. More precisely, we consider domains of the form 10.1.1 $$ G = \mathbb{D}\backslash \bigcup\limits_{j = 1}^N {\gamma _{j,} } $$ where γ j are analytic arcs satisfying certain technical conditions. They are the following (see Figure 10.1 for an example):

For a $$ \mathbb{D} $$ function ϑ, form the subspace $$ K_{z\vartheta } : = H^2 \left( \mathbb{D} \right) \cap (z\vartheta H^2 (\mathbb{D}))^ \bot $$ Since z ϑ H2 $$ \left( \mathbb{D} \right) $$ is an S-invariant subspace of H2 $$ \left( \mathbb{D} \right) $$ , then Kzϑ will be an S*-invariant subspace of H2 $$ \left( \mathbb{D} \right) $$ , where $$ S^* f = \frac{{f - f(0)}} {z} $$ is the backward shift operator. It is also easy to see that Kzϑ contains the constants. In fact, by Beurling’s theorem, every S*-invariant subspace, which also contains the constants. takes the form Kzϑ for some $$ \mathbb{D} $$ -inner function ϑ. It is well known [16, 26] that functions in Kzϑ have special ‘continuation’ properties. Indeed, recall from (3.3.2) that for h∈L1(m) $$ (Ch)(\lambda ) : = \int_\mathbb{T} {\frac{{h(\zeta )}} {{\zeta - \lambda }}dm} (\zeta ) $$ denotes the Cauchy transform of h. It is known [16, p. 87] that for any f∈Kzϑ the meromorphic function 4.1.1 $$ \tilde f(\lambda ) : = \frac{{C(f\overline {\zeta \vartheta } )(\lambda )}} {{C(\overline {\zeta \vartheta } )(\lambda )}} $$ on $$ \mathbb{D}_e $$ is a pseudocontinuation of f in that the non-tangential limits of f (from $$ \mathbb{D} $$ ) and $$ \tilde f $$ (from $$ \mathbb{D}_e $$ ) are equal almost everywhere on $$ \mathbb{T} $$ . Using the Cauchy integral formula and power series, one can prove the identity $$ \tilde f(\lambda ) = \frac{1} {{\vartheta ^* (\lambda )}}\sum\limits_{n = 1}^\infty {\frac{1} {{\lambda ^{n - 1} }}} \widehat{f\overline {\zeta \vartheta } } ( - n), $$ where $$ \hat \cdot (k) $$ denotes the k-th Fourier coefficient and 4.1.2 $$ \vartheta ^* (\lambda ): = \overline {\vartheta \left( {\begin{array}{*{20}c} 1 \\ {\overline{\overline \lambda } } \\ \end{array} } \right), } \lambda \in \mathbb{D}_e . $$ This says that 4.1.3 $$ \tilde f \in \frac{1} {{\vartheta ^* }}H^2 (\mathbb{D}_e ) \forall f \in K_{z\vartheta } . $$ .

We now examine the compression of S to certain co-invariant subspaces. Throughout this section ω will denote the harmonic measure for ∂G at some point in G. Note from (6.2.5) that $$ d\omega \asymp |\xi |^{ - 1/2} |\xi - 1|ds $$ . We begin with the following.

If M is an invariant subspace of H2 (G), the proof of Corollary 6.1.6. shows that $$ \mathcal{N}: = C_{\alpha ^{ - 1} } \circ \mathcal{M}$$ is a nearly invariant subspace of . We know from Corollary 3.2.9 that if {0, ∞} is not a subset of the common zeros of N and Φ and Ψ are the normalized reproducing kernels at z=0 and z=∞, then the smallest nearly invariant subspace containing Φ and Ψ is equal to N. From Remark 6.2.12 we also see that Φ ○ α is the normalized reproducing kernel for M at α1(0) while Ψ ○ α is the normalized reproducing kernel at α−1(∞).

If A is an irreducible essentially normal operator, then we prove that the C -algebra generated by A has a nite number of irreducible sub- normal operators as generators if and only if the essential spectrum of A is uncountable. It is shown that, in general, at most eight irreducible subnor- mal generators are required. Additionally, it is shown that frequently two irreducible subnormal operators will suce and that, in many instances, the subnormal operators can be taken to be unilateral shifts of multiplicity one or unitarily equivalent to the dual of the Bergman shift.