In this paper, we study the cyclicity of the shift operator $S$ acting on a Banach space $\mathcal {X}$ of analytic functions on the open unit disc $\mathbb {D}$. We develop a general framework where a method based on a corona theorem can be used to show that if $f,g\in \mathcal {X}$ satisfy $|g(z)|\leq |f(z)|$, for every $z\in \mathbb {D}$, and if g is cyclic, then f is cyclic. We also give sufficient conditions for cyclicity in this context. This enable us to recapture some recent results obtained in de Branges–Rovnayk spaces, in Besov–Dirichlet spaces and in weighted Dirichlet type spaces.

McCullough and Trent generalize Beurling–Lax–Halmos invariant subspace theorem for the shift on Hardy space of the unit disk to the multi-shift on Drury–Arveson space of the unit ball by representing an invariant subspace of the multi-shift as the range of a multiplication operator that is a partial isometry. By using their method, we obtain similar representations for a class of invariant subspaces of the multi-shifts on Hardy and Bergman spaces of the unit ball or polydisk. Our results are surprisingly general and include several important classes of invariant subspaces on the unit ball or polydisk.

This article describes Hilbert spaces contractively contained in certain reproducing kernel Hilbert spaces of analytic functions on the open unit disc which are nearly invariant under division by an inner function. We extend Hitt’s theorem on nearly invariant subspaces of the backward shift operator on $H^2(\mathbb {D})$ as well as its many generalizations to the setting of de Branges spaces.

We show that properties of pairs of finite, positive, and regular Borel measures on the complex unit circle such as domination, absolute continuity, and singularity can be completely described in terms of containment and intersection of their reproducing kernel Hilbert spaces of “Cauchy transforms” in the complex unit disk. This leads to a new construction of the classical Lebesgue decomposition and proof of the Radon–Nikodym theorem using reproducing kernel theory and functional analysis.

On all Bergman–Besov Hilbert spaces on the unit disk, we find self-adjoint weighted shift operators that are differential operators of half-order whose commutators are the identity, thereby obtaining uncertainty relations in these spaces. We also obtain joint average uncertainty relations for pairs of commuting tuples of operators on the same spaces defined on the unit ball. We further identify functions that yield equality in some uncertainty inequalities.

We characterize the membership in the Schatten ideals $\mathcal {S}_p$, $0<p<\infty $, of composition operators acting on weighted Dirichlet spaces. Our results concern a large class of weights. In particular, we examine the case of perturbed superharmonic weights. Characterization of composition operators acting on weighted Bergman spaces to be in $\mathcal {S}_p$ is also given.

Motivated by the near invariance of model spaces for the backward shift, we introduce a general notion of $(X,Y)$-invariant operators. The relations between this class of operators and the near invariance properties of their kernels are studied. Those lead to orthogonal decompositions for the kernels, which generalize well-known orthogonal decompositions of model spaces. Necessary and sufficient conditions for those kernels to be nearly X-invariant are established. This general approach can be applied to a wide class of operators defined as compressions of multiplication operators, in particular to Toeplitz operators and truncated Toeplitz operators, to study the invariance properties of their kernels (general Toeplitz kernels).

We investigate unbounded, linear operators arising from a finite sum of composition operators on Fock space. Real symmetry and complex symmetry of these operators are characterised.

We show that any weakly separated Bessel system of model spaces in the Hardy space on the unit disc is a Riesz system and we highlight some applications to interpolating sequences of matrices. This will be done without using the recent solution of the Feichtinger conjecture, whose natural generalization to multidimensional model subspaces of ${\mathrm {H}}^2$ turns out to be false.

Let $\left( X,\,B,\,\mu \right)$ be a $\sigma $-finite measure space and let $H\,\subset \,{{L}^{2}}\left( X,\,\mu \right)$ be a separable reproducing kernel Hilbert space on $X$. We show that the multiplier algebra of $H$ has property $\left( {{A}_{1}}\left( 1 \right) \right)$.

We study the Friedrichs extensions of unbounded cyclic subnormals. The main result of the present paper is the identification of the Friedrichs extensions of certain cyclic subnormals with their closures. This generalizes as well as complements the main result obtained in [5]. Such characterizations lead to abstract Galerkin approximations, generalized wave equations, and bounded -functional calculi.

We combine the theory of sectorial sesquilinear forms with the theory of unbounded subnormal operators in Hilbert spaces to characterize the Friedrichs extensions of multiplication operators (with analytic symbols) in certain functional Hilbert spaces. Such characterizations lead to abstract Galerkin approximations and generalized wave equations.