Vol 52, No 1 (2018)

We introduce Duhamel algebras and study their properties and applications. We prove that a Banach space of analytic functions on the unit disc that satisfy certain conditions is a Duhamel algebra and describe its closed ideals. These results substantially generalize and improve the main results of Wigley’s papers. Some other related questions are also discussed.

Let M be a subharmonic function with Riesz measure ν_M in a domain D in the n-dimensional complex Euclidean space ℂ_n, and let f be a nonzero function that is holomorphic in D, vanishes on a set Z ⊂ D, and satisfies |f| ⩽ expM on D. Then restrictions on the growth of ν_M near the boundary of D imply certain restrictions on the dimensions or the area/volume of Z. We give a quantitative study of this phenomenon in the subharmonic framework.

Analogues of the Pólya–Szégö inequality with variable exponent in the integrand are considered. Necessary and sufficient conditions for the fulfillment of these inequalities are obtained.

It is proved that the commutative algebra A of operators on a reflexive real Banach space has an invariant subspace if each operator T ∈ A satisfies the condition

where ║ · ║_e denotes the essential norm. This implies the existence of an invariant subspace for any commutative family of essentially self-adjoint operators on a real Hilbert space.

It is shown that, for any compact set K ⊂ ℝⁿ (n ⩾ 2) of positive Lebesgue measure and any bounded domain G ⊃ K, there exists a function in the Hölder class C^1,1(G) that is a solution of the minimal surface equation in G \ K and cannot be extended from G \ K to G as a solution of this equation.

Spectral asymptotics of the Sturm–Liouville problem with an arithmetically self-similar singular weight is considered. Previous results by A. A. Vladimirov and I. A. Sheipak, and also by the author, rely on the spectral periodicity property, which imposes significant restrictions on the self-similarity parameters of the weight. This work introduces a new method for estimating the eigenvalue counting function. This makes it possible to consider a much wider class of self-similar measures.

We consider a system of two first-order difference equations in the complex plane. We assume that the matrix of the system is a 1-periodic meromorphic function having two simple poles per period and bounded as Im z → ±∞. We prove the existence and uniqueness of minimal meromorphic solutions, i.e., solutions having simultaneously a minimal set of poles and minimal possible growth as Im z → ±∞. We consider the monodromy matrix representing the shift-byperiod operator in the space of meromorphic solutions and corresponding to a basis built of two minimal solutions. We check that it has the same functional structure as the matrix of the initial system of equations and, in particular, is a meromorphic periodic function with two simple poles per period. This implies that the initial equation is invariant with respect to the monodromization procedure, that is, a natural renormalization procedure arising when trying to extend the Floquet–Bloch theory to difference equations defined on the real line or complex plane and having periodic coefficients. Our initial system itself arises after one renormalization of a self-adjoint difference Schrödinger equation with 1-periodic meromorphic potential bounded at ±i∞ and having two poles per period.