We consider Hilbert spaces of analytic functions on a plane domain Ω and multiplication operators on such spaces induced by functions from $H^{\infty }\left(\Omega \right)$. Recently, K. Zhu has given conditions under which the adjoints of multiplication operators on Hilbert spaces of analytic functions belong to the Cowen-Douglas classes. In this paper, we provide some sufficient conditions which give the converse of the main result obtained by K. Zhu. We also characterize the commutant of certain multiplication operators.

For an arbitrary set $A\subseteq {\mathbb{N}}^{+}$ containing 1, an ergodic automorphism T whose set of essential values of the multiplicity function is equal to A is constructed. If A is additionally finite, T can be chosen to be an analytic diffeomorphism on a finite-dimensional torus.

Some general multiplicity results for critical points of parameterized functionals on reflexive Banach spaces are established. In particular, one of them improves some aspects of a recent result by B. Ricceri. Applications to boundary value problems are also given.

Recent papers have studied the existence of phase transition solutions for Allen–Cahn type equations. These solutions are either single or multi-transition spatial heteroclinics or homoclinics between simpler equilibrium states. A sufficient condition for the construction of the multitransition solutions is that there are gaps in the ordered set of single transition solutions. In this paper we explore the necessity of these gap conditions.

We study a class of logarithmic fractional Schrödinger equations with possibly vanishing potentials. By using the fibrering maps and the Nehari manifold we obtain the existence of at least one nontrivial solution.

Let $\Gamma$ be a rectifiable Jordan curve in the finite complex plane $ℂ$ which is regular in the sense of Ahlfors and David. Denote by $L_C^2\left(\Gamma \right)$ the space of all complex-valued functions on $\Gamma$ which are square integrable w.r. to the arc-length on $\Gamma$. Let $L^2\left(\Gamma \right)$ stand for the space of all real-valued functions in $L_C^2\left(\Gamma \right)$ and put $$L_0^2\left(\Gamma \right)=\{h\in L^2\left(\Gamma \right){\int }_{\Gamma }h\left(\zeta \right)\left|\mathrm{d}\zeta \right|=0\}.$$ Since the Cauchy singular operator is bounded on $L_C^2\left(\Gamma \right)$, the Neumann-Poincaré operator $C_1^{\Gamma }$ sending each $h\in L^2\left(\Gamma \right)$ into $$C_1^{\Gamma }h\left({\zeta }_0\right):=\Re {\left(\pi \mathrm{i}\right)}^{-1}\mathrm{P}.V.{\int }_{\Gamma }\frac{h\left(\zeta \right)}{\zeta -{\zeta }_0}\mathrm{d}\zeta ,{\zeta }_0\in \Gamma ,$$ is bounded on $L^2\left(\Gamma \right)$. We show that the inclusion $$C_1^{\Gamma }\left(L_0^2\left(\Gamma \right)\right)\subset L_0^2\left(\Gamma \right)$$ characterizes the circle in the class of all...

Using a weaker version of the Newton-Kantorovich theorem, we provide a discretization result to find finite element solutions of elliptic boundary value problems. Our hypotheses are weaker and under the same computational cost lead to finer estimates on the distances involved and a more precise information on the location of the solution than before.

We consider the Neumann problem for the equation $-\Delta u-\lambda u={Q\left(x\right)\left|u\right|}^{2*-2}u$, u ∈ H¹(Ω), where Q is a positive and continuous coefficient on Ω̅ and λ is a parameter between two consecutive eigenvalues ${\lambda }_{k-1}$ and ${\lambda }_k$. Applying a min-max principle based on topological linking we prove the existence of a solution.

In this paper, we develop the approach and techniques of [Boucherif A., Precup R., Semilinear evolution equations with nonlocal initial conditions, Dynam. Systems Appl., 2007, 16(3), 507–516], [Zhou Y., Jiao F., Nonlocal Cauchy problem for fractional evolution equations, Nonlinar Anal. Real World Appl., 2010, 11(5), 4465–4475] to deal with nonlocal Cauchy problem for semilinear fractional order evolution equations. We present two new sufficient conditions on existence of mild solutions. The first...

Let X and Y be Banach spaces and let 𝓐(X,Y) be a closed subspace of 𝓛(X,Y), the Banach space of bounded linear operators from X to Y, containing the subspace 𝒦(X,Y) of compact operators. We prove that if Y has the metric compact approximation property and a certain geometric property M*(a,B,c), where a,c ≥ 0 and B is a compact set of scalars (Kalton's property (M*) = M*(1, {-1}, 1)), and if 𝓐(X,Y) ≠ 𝒦(X,Y), then there is no projection from 𝓐(X,Y) onto 𝒦(X,Y) with norm less than max|B| + c....

Let A and B be two -non necessarily bounded- normal operators. We give new conditions making their product normal. We also generalize a result by Deutsch et al on normal products of matrices.

Let T be a bounded linear operator acting on a complex, separable, infinite-dimensional Hilbert space and let f: D → ℂ be an analytic function defined on an open set D ⊆ ℂ which contains the spectrum of T. If T is the limit of hypercyclic operators and if f is nonconstant on every connected component of D, then f(T) is the limit of hypercyclic operators if and only if $f\left({\sigma }_W\left(T\right)\right)\cup z\in ℂ:\left|z\right|=1$ is connected, where ${\sigma }_W\left(T\right)$ denotes the Weyl spectrum of T.