Let X,Y be realcompact spaces or completely regular spaces consisting of Gδ-points. Let φ be a linear bijective map from C(X) (resp. C(X)) onto C(Y ) (resp. C(Y )). We show that if φ preserves nonvanishing functions, that is, f(x) 6= 0,∀x ∈ X, ⇐⇒ φ(f)(y) 6= 0,∀ y ∈ Y, then φ is a weighted composition operator φ(f) = φ(1) · f ◦ τ, arising from a homeomorphism τ : Y → X. This result is applied al...

The study of the point spectrum and the singular continuous one is reduced to investigating the structure of the real roots set of an analytic function with positive imaginary partM(λ). We prove a uniqueness theorem for such a class of analytic functions. Combining this theorem with a lemma on smoothness of M(λ) near its real roots permits us to describe the density of the singular spectrum. 20...

Recommended by Shusen Ding Let D denote the open unit disk in the complex plane and let dAz denote the normalized area measure on D. Φ α is defined as follows L Φ α {f ∈ HD : D ΦΦlog |fz|1 − |z| 2 α dAz < ∞}. Let ϕ be an analytic self-map of D. The composition operator C ϕ induced by ϕ is defined by C ϕ f f • ϕ for f analytic in D. We prove that the composition operator C ϕ is compact on L Φ α ...

with Dirichlet boundary conditions, acting on l2(N,C2), resp. L2([0,∞),C2), where c > 0 represents the speed of light, m ≥ 0 the mass of a particle, I2 is the 2× 2 identity matrix and V is a bounded real potential. In the discrete case D is the finite difference operator defined by (Dφ)(n) = φ(n+1)−φ(n), with adjoint (Dφ)(n) = φ(n − 1) − φ(n), and in the continuous case D = D = −i d dx . Model ...

whenever the right-hand side of (1.2) represents an analytic function in a neighborhood of the origin. When φ(x) is an entire function, the operator φ(D) has been studied by several authors (see, for example, [5, §11], [19, Chapter IX], [22] and [32]). The conjecture of Pólya and Wiman, proved in [8], [9] and [17], states that if f(x) ∈ L-P∗, then Df(x) is in the Laguerre-Pólya class for all su...

An analytic self-map φ : D → D of the open unit disk D in the complex plane induces the composition operator Cφ on H(D), the space of holomorphic functions on D, defined by Cφ(f) = f ◦φ. A basic goal in the study of composition operators is to relate function theoretic properties of φ to operator theoretic properties of Cφ. Here we review some results that characterize when φ induces a bounded ...

The invariant subspace problem relative to a von Neumann algebra M ⊆ B(H) asks whether every operator T ∈ M has a proper, nontrivial invariant subspace H0 ⊆ H such that the orthogonal projection p onto H0 is an element of M; equivalently, it asks whether there is a projection p ∈ M, p / ∈ {0, 1}, such that Tp = pTp. Even when M is a II1–factor, this invariant subspace problem remains open. In t...

In our previous study, we defined the semantics of modal μ-calculus on minplus algebra N∞ and developed a model-checking algorithm. N∞ is the set of all natural numbers and infinity (∞), and has two operations min and plus. In the semantics, disjunctions are interpreted by min and conjunctions by plus. This semantics allows interesting properties of a Kripke structure, such as the shortest path...

We take an analytic approach to the CFT bootstrap, studying the 4-pt correlators of d > 2 dimensional CFTs in an Eikonal-type limit, where the conformal cross ratios satisfy |u| |v| < 1. We prove that every CFT with a scalar operator φ must contain infinite sequences of operators Oτ,` with twist approaching τ → 2∆φ + 2n for each integer n as `→∞. We show how the rate of approach is controlled b...