The weighted Weiss conjecture states that the system theoretic property of weighted admissibility can be characterised by a resolvent growth condition. For positive weights, it is known that the conjecture is true if the system is governed by a normal operator; however, the conjecture fails if the system operator is the unilateral shift on the Hardy space H 2 (D) (discrete time) or the right-shift semigroup on L 2 (R + ) (continuous time). To contrast and complement these counterexamples, in this paper positive results are presented characterising weighted admissibility of linear systems governed by shift operators and shift semigroups. These results are shown to be equivalent to the question of whether certain generalized Hankel operators satisfy a reproducing kernel thesis.

We consider cyclic m‐isometries on a complex separable Hilbert space. Such operators are characterized in terms of shifts on abstract spaces of weighted Dirichlet type. Our results resemble those of Agler and Stankus, but our model spaces are described in terms of Dirichlet integrals rather than analytic Dirichlet operators. The chosen point of view allows us to construct a variety of examples. An interesting feature among all of these is that the corresponding model spaces are contained in a certain subspace of the Hardy space H2, depending only on the order of the corresponding operator. We also demonstrate how our framework allows for the construction of unbounded m‐isometries.

Abstract. Let BMOA N P (L) denote the space of L-valued analytic functions φ for which the Hankel operator Γ φ is H 2 (H)-bounded. Obtaining concrete characterizations of BMOA N P (L) has proven to be notoriously hard. Let D α denote fractional differentiation. Motivated originally by control theory, we characterize H 2 (H)-boundedness of D α Γ φ , where α > 0, in terms of a natural anti-analytic Carleson embedding condition. We obtain three notable corollaries: The first is that BMOA N P (L) is not characterized by said embedding condition. The second is that when we add an adjoint embedding condition, we obtain a sufficient but not necessary condition for boundedness of Γ φ . The third is that there exists a bounded analytic function for which the associated anti-analytic Carleson embedding is unbounded. As a consequence, boundedness of an analytic Carleson embedding does not imply that the anti-analytic ditto is bounded. This answers a question by Nazarov, Pisier, Treil, and Volberg. IntroductionThroughout this paper we let H denote a separable Hilbert space with inner product ·, · H . Unless we explicitly state otherwise, we assume that H is infinite-dimensional. We denote by L = L (H) the space of bounded linear transformations on H, by S 1 the corresponding trace class, and by S 2 the Hilbert-Schmidt class. X will be used as a generic notation for an element of the set H, L, S 1 , S 2 . We will use Y to denote a general Banach space. By Hol (Y) we denote the space of Y-valued analytic functions on the open unit disc D. For f ∈ Hol(Y), we denote the nth Taylor coefficients at the origin byf (n). We denote by O (Y) the space of functions in Hol (Y) that admit an analytic extension to a larger disc (centered at the origin). If Y = C, then we suppress this in our notation, i.e. Hol = Hol (C), and O = O (C). The same principle will apply to all function spaces discussed below.For p ∈ [1, ∞] and X ∈ H, S 1 , we let L p (T, X ) denote the standard space of p-Bochner-Lebesgue integrable functions from T to X . Here T denotes the unit circle inHere m denotes normalized Lebesgue measure on T. E. RYDHE GAFAThe Hardy space H p (X ) is the space of f ∈ Hol (X ) such thatwhere we have defined the function f r : z → f (rz). An important property of Hardy space functions is that they have boundary values in a natural sense, cf. Proposition 2.1. We denote the boundary values of f ∈ H p (X ) by bf ∈ L p (T, X ). The space H 2 (H) is a Hilbert space, with inner product f, g = ∞ 0f (n), g(n) H . Of particular importance will be the set of H 2 (H)-normalized functions in O (H), which we denote by O 1 (H).We now introduce the main topics of this paper. Initially, we consider the scalar setting, rather than the proper vectorial one. Hankel operators.Given φ ∈ Hol and f ∈ O, we define the action of the Hankel operator Γ φ on f byA standard reference on Hankel operators is [Pel03]. We refer to φ as the symbol of Γ φ . We say that Γ φ is bounded if it extends to a bounded operator on H 2 . For Γ φ to be bounded it is necessary for ...

We investigate suitable conditions for a C0‐semigroup false(T(t)false)t⩾0 of Hilbert space contractions to be unitarily equivalent to the restriction of the adjoint shift semigroup false(Sγ*(t)false)t⩾0 to an invariant subspace of the standard weighted Bergman space Aγ−2false(C+,scriptKfalse). It turns out that false(T(t)false)t⩾0 admits a model by false(Sγ*(t)false)t⩾0 if and only if its cogenerator is γ‐hypercontractive and trueprefixlimt→0Tfalse(tfalse)=0 in strong operator topology. We then discuss how such semigroups can be characterized without involving the cogenerator. A sufficient condition is that, for each t>0, the operator T(t) is γ‐hypercontractive. Surprisingly, this condition is necessary if and only if γ is integer. The paper is concluded with a conjecture that would imply a more symmetric characterization.

We investigate so-called Laplace-Carleson embeddings for large exponents. In particular, we extend some results by Jacob, Partington, and Pott. We also discuss some related results for Sobolev-and Besov spaces, and mapping properties of the Fourier transform. These variants of the Hausdorff-Young theorem appear difficult to find in the literature. We conclude the paper with an example related to an open problem.