Complex Analysis and Operator Theory最新文献

The main theorem of this article is a Runge type theorem proved for k-differentials ((kge 2)). The integrability in the (L^1)- norm is defined for k-differentials in Section 2. We consider k-differentials which are integrable in the defined (L^1)- norm on the Riemann surface, and are holomorphic on an open subset of that surface. We will show those k-differentials can be approximated by meromorphic k-differentials. The proof applies a generalized form of the Poincaré series map. This generalized form is proved in Section 3. Section 2 contains the definition of the Poincaré series and its convergence, with particular focus on the convergence of the Poincaré series for rational functions, which is applied in the main theorem. Sections 3 and 4 contain the new results proved in this paper. The statement and proof of the main theorem are in Section 4.

Abstract

Motivated by Liu’s (Sci China Math 66:1199–1216, 2023) recent work. This article reveals the essential features of Hahn polynomials by presenting a new q-exponential operator, that is $$begin{aligned} exp _q(tDelta _{x,a})f(x)=frac{(axt;q)_{infty }}{(xt;q)_{infty }} sum _{n=0}^{infty }frac{t^n}{(q;q)_n} f(q^n x) end{aligned}$$ with (Delta _{x,a}=x (1-a)eta _a+eta _x) and (eta _x {f(x) }=f(qx)) . Letting (f(x) equiv 1) and the above operator equation immediately becomes the generating function of Hahn polynomials. These lead us to use a systematic method for studying identities involving Hahn polynomials. As applications, we use the method of the q-exponential operator to prove some new q-identities, including q-Nielsen’s formulas and Carlitz’s extension for the Hahn polynomials, etc. Moreover, a generalization of q-Gauss summation is given, too.

For a submodule ({mathcal {M}}) in Hardy module (H^2({mathbb {D}}^n)) on the unit polydisc in (mathbb {C}^{n}), we define the (n-1) tuple of fringe operators (textbf{F}=(F_{1},F_{2},ldots ,F_{n-1})) and the n tuple of restriction operators (textbf{R}=(R_{z_{1}},R_{z_{2}},ldots , R_{z_{n}})) with respect to ({mathcal {M}}). In this paper, for the case (n=3), it is shown that the fringe operators (textbf{F}) are Fredholm if and only if the tuple (textbf{R}-lambda ) is Fredholm, where (lambda in {mathbb {D}}^3), and moreover (ind(textbf{F})=-ind(mathbf{R-lambda })), which answer a question of Yang (Proc Am Math Soc 131 (2):533–541, 2003) partly and generalize a result of Luo et al. (J Math Anal Appl 465(1):531–546, 2018) in the case (n=2). Finally, we also discuss the difference quotient operators in (H^2({mathbb {D}}^n)), and apply them to explore the relationship between the fringe operators and compression operators on quotient module.

What structure can be placed on the burgeoning field of fractional calculus with assorted kernel functions? This question has been addressed by the introduction of various general kernels, none of which has both a fractional order parameter and a clear inversion relation. Here, we use ideas from abstract algebra to construct families of fractional integral and derivative operators, parametrised by a real or complex variable playing the role of the order. These have the typical behaviour expected of fractional calculus operators, such as semigroup and inversion relations, which allow fractional differential equations to be solved using operational calculus in this general setting, including all types of fractional calculus with semigroup properties as special cases.

In this paper, we are concerned with the existence of least energy sign-changing solutions for the following N-Laplacian Kirchhoff-type problem with logarithmic and exponential nonlinearities:

where f(t) behaves like ( expleft( {alpha |t|^{{frac{N}{{N - 1}}}} } right) ). Combining constrained variational method, topological degree theory and quantitative deformation lemma, we prove that the problem possesses one least energy sign-changing solution (u_{b}) with precisely two nodal domains. Moreover, we show that the energy of (u_{b}) is strictly larger than two times of the ground state energy and analyze the convergence property of (u_{b}) as (bsearrow 0).

Let A and E be self-adjoint matrices or operators on (ell ^2({{mathbb {N}}})), where A is fixed and E is a small perturbation. We study how the eigenvalues of (A+E) depend on E, with the aim of obtaining second order formulas that are explicitly computable in terms of the spectral decomposition of A and a certain block decomposition of E. In particular we extend the classical Rayleigh-Schrödinger formulas for the one-parameter perturbation (A+tE) where (tin {{mathbb {R}}}) varies and E is held fixed, by dropping t and considering E as the variable.

Abstract

For the quaternion algebra ({mathbb {H}}) and (f:mathbb R^2rightarrow {mathbb {H}}) , we consider a two-sided quaternion Fourier transform (widehat{f}) . Necessary and sufficient conditions for f to belong to generalized uniform Lipschitz spaces are given in terms of behavior of (widehat{f}) .

Abstract

We investigate some new classes of operator algebras which we call semi- (sigma ) -finite subdiagonal and Riesz approximable. These constitute the most general setting to date for a noncommutative Hardy space theory based on Arveson’s subdiagonal algebras. We develop this theory and study the properties of these new classes.

In this paper we characterize completely the septuple

such that the fractional integration operator ({mathfrak {I}}_t), of order (t in {mathbb {C}}), is bounded between two mixed norm spaces:

We treat three types of definitions for ({mathfrak {I}}_t): Hadamard, Flett, and Riemann-Liouville. Our main result (Theorem 2) extends that of Buckley-Koskela-Vukotić in 1999 on the Bergman spaces (Theorem B), and the case (t=0) recovers the embedding theorem of Arévalo in 2015 (Corollary 3). The corresponding result for the Hardy spaces (H^p({mathbb {D}})), of type Riemann-Liouville, is due to Hardy and Littlewood in 1932.

We study the algebra generated by all Toeplitz operators on the Fréchet space of all entire functions and all functions holomorphic in the unit disk. In both cases we prove that the quotient algebra by the commutator ideal can be equipped with a locally convex topology which makes this quotient algebra algebraically and topologically isomorphic with the symbol algebra. We also show that the topology of uniform convergence on bounded sets is not a correct choice here, since the commutator ideal is dense in the algebra generated by all Toeplitz operators in this topology. However, the linear space of all Toeplitz operators with the topology of uniform convergence on bounded sets is linearly isomorphic with the symbol space. There are also some other subtle differences between the case which we study and the classical one. Our theorems provide the key step towards extending the results previously obtained for single Toeplitz operators to the elements of the algebra generated by all Toeplitz operators.