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We generalize the theory of homotopic deviation of square complex matrices to regular matrix pencils. To this end, we study the existence and the analyticity of the resolvent of the matrix pencils whose matrices are under homotopic deviation with the deviation parameter $ t \in \mathbb {C} $ t ∈ C . Moreover, we investigate and identify the limits of both the resolvent and the spectrum of the deviated matrix pencils, as $ | t | \to \infty $ | t | → ∞. We also study the special cases where t tends to the eigenvalues of the related matrix pairs. We use the notions and the results of the generalized homotopic deviation theory to analyze the Weierstrass structure of the deviated matrix pencils under two different assumptions, in particular, either the eigenvalues of the deviated matrix pencils are independent parameters, or the deviation parameter t is an independent parameter. Numerical examples illustrate and support the theoretical results. [ABSTRACT FROM AUTHOR]

We consider a Sturm-Liouville-type differential operator defined on a separable Hilbert space with bounded operator coefficient and periodic boundary conditions given on a finite interval. We obtain an explicit formula which we call the second regularized trace formula of the mentioned operator. [ABSTRACT FROM AUTHOR]

In this work, we study the solvability and maximal regularity questions for the singular third‐order differential equation with unbounded coefficients and some applications. Unlike previously studied cases, the leading and intermediate coefficients of this equation can grow independently. We obtain sufficient coefficient conditions for correctness of this equation and compactness for inverse of the corresponding differential operator. We also prove the maximal regularity estimate for a generalized solution. Using these results, we obtain upper and lower estimates for the number of Kolmogorov k‐diameters of the set associated with the linear Korteweg–de Vries equation's solutions. We give an example and show that from the above inequalities follow two‐sided estimates of the Kolmogorov k‐diameters. [ABSTRACT FROM AUTHOR]

The determination of a Galois group is an important question in computational algebraic number theory. One approach is based on the inspection of resolvents. This article reports on this method and on the performance of the current magma [W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput., 24 (1997) 235-265]. implementation. [ABSTRACT FROM AUTHOR]

In this paper, we present a technique for constructing generalized solutions of the Cauchy problem for abstract integro-differential equations with degeneration in Banach spaces. A generalized solution is constructed as the convolution of the fundamental operator function (fundamental solution, influence function) of the integro-differential operator of the equation with a generalized function of a special form, which involves all input data of the original problem. Based on the analysis of the representation for the generalized solution, we obtain sufficient solvability conditions for the original Cauchy problem in the class of functions of finite smoothness. Under these sufficient conditions, the generalized solution constructed turns out to be a classical solution with the required smoothness. The abstract results obtained in the paper are applied to the study of applied initial-boundary-value problems from the theory of oscillations in viscoelastic media. [ABSTRACT FROM AUTHOR]

In this paper, we investigate and analyse the strong convergence of the sequence generated by an inexact proximal point method with possible unbounded errors for finding zeros of monotone operators in Hadamard spaces. We show that the boundedness of the generated sequence is equivalent to the zero set of the operator to be nonempty. In this case, we prove the strong convergence of the generated sequence to a zero of the operator. We also provide some applications of our main results and give a numerical example to show the performance of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Let Ω a bounded domain in Rn with smooth boundary ∂Ω. In this article, in view of our earlier paper [12], let the conditions made on this paper be sufficiently more general than [12]. We investigate the spectral properties of a non-selfadjoint elliptic differential operator (Pu)(x) = -Σni,j=1 (ρ2α(x)aij(x)Q(x)u'xi(x))'xj, acting on Hilbert space H∂ = L²(Ω)∂. Here c|s|² ≤ Σni,j=1 aij(x)sisj (s = (s1,. . ., sn) ∊ Cn, x ∊ Ω), ρ(x) = dist{x, ∂Ω}, aij(x) = aji(x) ∊ C²(Ω), 0 ≤ α < Ω. Fur-thermore, suppose that Q(x) ∊ C²(Ω, End C∂) such that for each x ∊ Ω the matrix function Q(x) has non-zero simple eigenvalues μj (x) ∊ C²(Ω) (1 ∊ j ℹ ∂) lie in the C\ɸ, where ɸ = {z ⅂ C: |arg z| ∊ φ}, φ 2 (0, π). [ABSTRACT FROM AUTHOR]

For any pseudo-Riemannian hyperbolic space X over R , C , H or O , we show that the resolvent R (z) = (□ − z Id) − 1 of the Laplace–Beltrami operator −□ on X can be extended meromorphically across the spectrum of □ as a family of operators C c ∞ (X) → D ′ (X). Its poles are called resonances and we determine them explicitly in all cases. For each resonance, the image of the corresponding residue operator in D ′ (X) forms a representation of the isometry group of X , which we identify with a subrepresentation of a degenerate principal series. Our study includes in particular the case of even functions on de Sitter and Anti-de Sitter spaces. For Riemannian symmetric spaces analogous results were obtained by Miatello–Will and Hilgert–Pasquale. The main qualitative differences between the Riemannian and the non-Riemannian setting are that for non-Riemannian spaces the resolvent can have poles of order two, it can have a pole at the branching point of the covering to which R (z) extends, and the residue representations can be infinite-dimensional. [ABSTRACT FROM AUTHOR]

For the Frobenius matrix accompanying an algebraic (differential) equation in a complex Banach algebra, the Cayley–Hamilton theorem is proved, which is used to obtain a representation of the resolvent. [ABSTRACT FROM AUTHOR]

Finding a zero of a sum of maximally monotone operators is a fundamental problem in modern optimization and nonsmooth analysis. Assuming that the resolvents of the operators are available, this problem can be tackled with the Douglas-Rachford algorithm. However, when dealing with three or more operators, one must work in a product space with as many factors as there are operators. In groundbreaking recent work by Ryu and by Malitsky and Tam, it was shown that the number of factors can be reduced by one. A similar reduction was achieved recently by Campoy through a clever reformulation originally proposed by Kruger. All three splitting methods guarantee weak convergence to some solution of the underlying sum problem; strong convergence holds in the presence of uniform monotonicity. In this paper, we provide a case study when the operators involved are normal cone operators of subspaces and the solution set is thus the intersection of the subspaces. Even though these operators lack strict convexity, we show that striking conclusions are available in this case: strong (instead of weak) convergence and the solution obtained is (not arbitrary but) the projection onto the intersection. To illustrate our results, we also perform numerical experiments. [ABSTRACT FROM AUTHOR]

This paper is concerned with a mixed type differential operator Lu=kyuxx−uyy+byux+qyu,$$ Lu=k(y){u}_{xx}-{u}_{yy}+b(y){u}_x+q(y)u, $$which is initially defined with C0,π∞Ω‾$$ {C}_{0,\pi}^{\infty}\left(\overline{\Omega}\right) $$, where Ω‾={x,y:−π≤x≤π,−∞

In this study, we consider the Oseen structure of the linearization of a compressible fluid–structure interaction (FSI) system for which the interaction interface is under the effect of material derivative term. The flow linearization is taken with respect to an arbitrary, variable ambient vector field. This process produces extra "convective derivative" and "material derivative" terms, which render the coupled system highly nondissipative. We show first a new well‐posedness result for the full incorporation of both Oseen terms, which provides a uniformly bounded semigroup via dissipativity and perturbation arguments. In addition, we analyze the long time dynamics in the sense of asymptotic (strong) stability in an invariant subspace (one‐dimensional less) of the entire state space, where the continuous semigroup is uniformly bounded. For this, we appeal to the pointwise resolvent condition introduced in Chill and Tomilov [Stability of operator semigroups: ideas and results, perspectives in operator theory Banach center publications, 75 (2007), Institute of Mathematics Polish Academy of Sciences, Warszawa, 71–109], which avoids an immensely technical and challenging spectral analysis and provides a short and relatively easy‐to‐follow proof. [ABSTRACT FROM AUTHOR]

We show that strongly bounded operators in locally convex spaces and topological vector spaces under some additional conditions have the spectral properties similar to those of bounded operators in normed spaces. Such operator enjoys the Berling–Gelfand formula in a sequentially complete locally convex space. [ABSTRACT FROM AUTHOR]

We generalize the theory of homotopic deviation of square complex matrices to regular matrix pencils. To this end, we study the existence and the analyticity of the resolvent of the matrix pencils whose matrices are under homotopic deviation with the deviation parameter t ∈ C . Moreover, we investigate and identify the limits of both the resolvent and the spectrum of the deviated matrix pencils, as | t | → ∞ . We also study the special cases where t tends to the eigenvalues of the related matrix pairs. We use the notions and the results of the generalized homotopic deviation theory to analyze the Weierstrass structure of the deviated matrix pencils under two different assumptions, in particular, either the eigenvalues of the deviated matrix pencils are independent parameters, or the deviation parameter t is an independent parameter. Numerical examples illustrate and support the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we study a Yosida variational inclusion problem with its corresponding Yosida resolvent equation problem. We mention some schemes to solve both the problems, but we focus our study on discussing convergence criteria for the Yosida variational inclusion problem in real Banach space and for the Yosida resolvent equation problem in q-uniformly smooth Banach space. For faster convergence, we apply an inertial extrapolation scheme for both the problems. An example is provided. [ABSTRACT FROM AUTHOR]

Topological insulators (TIs) are renowned for their remarkable electronic properties: quantized bulk Hall and edge conductivities, and robust edge wave-packet propagation, even in the presence of material defects and disorder. Computations of these physical properties generally rely on artificial periodicity (the supercell approximation, which struggles in the presence of edges), or unphysical boundary conditions (artificial truncation). In this work, we build on recently developed methods for computing spectral properties of infinite-dimensional operators. We apply these techniques to develop efficient and accurate computational tools for computing the physical properties of TIs. These tools completely avoid such artificial restrictions and allow one to probe the spectral properties of the infinite-dimensional operator directly, even in the presence of material defects, edges and disorder. Our methods permit computation of spectra, approximate eigenstates, spectral measures, spectral projections, transport properties and conductances. Numerical examples are given for the Haldane model, and the techniques can be extended similarly to other TIs in two and three dimensions. [ABSTRACT FROM AUTHOR]

For a bounded linear operator on a Banach space, we study approximation of the spectrum and pseudospectra in the Hausdorff distance. We give sufficient and necessary conditions in terms of pointwise convergence of appropriate spectral quantities. [ABSTRACT FROM AUTHOR]

Until now, the correspondence between the Alexander–Kolmogorov complex, and the de Rham one, by means of a small scale parameter, has not gone that far as passing to the limit of the resolvent of the associated Laplacian, when the small parameter tends towards zero. Along these lines, a result proving a complete Hodge decomposition was missing. We bridge this gap by means of our own rescaled h-cohomology, h being a very small parameter. Passing to the limit of the resolvent enables us to consider the extension to singular spaces, in particular, our h-differential operators also enable us to also make the connection with those of analysis on fractals, as introduced by Jun Kigami, and taken up by Robert S. Strichartz. [ABSTRACT FROM AUTHOR]

In this paper, we present a method for finding the zero of the sum of finite family of maximal monotone operators on real Hilbert spaces. In the case where the number of maximal monotone operators is three, we define a function such that its fixed points are solutions of our problem. Some illustrative examples are given at the end of this paper. [ABSTRACT FROM AUTHOR]

In this paper, we give three different new proofs of the validity of the geometry conjecture about cycles of projections onto nonempty closed, convex subsets of a Hilbert space. The first uses a simple minimax theorem, which depends on the finite dimensional Hahn-Banach theorem. The second uses Fan's inequality, which has found many applications in optimization and mathematical economics. The third uses three results on maximally monotone operators on a Hilbert space. [ABSTRACT FROM AUTHOR]

In this article, we study an extended split equality variational inclusion and fixed point problems which is an extension of the split equality variational inclusion problems and fixed point problems. We propose a simultaneous inertial type iterative algorithm with a self adaptive stepsizes such that there is no need for a prior information about the operator norm. We further stated and prove that the proposed algorithm weakly converges to a solution of the extended split equality variational inclusion and fixed point problems. Finally, we give some numerical examples to demonstrate the performance and the applicability of the proposed algorithm. The results of this paper complements and extends results on split equality variational inclusion and fixed point problems. [ABSTRACT FROM AUTHOR]

Bhargava parametrized quintic rings over Z by quadruples of 5 × 5 alternating matrices. We extend the construction to work similarly over any Dedekind domain R. No assumptions are needed on the characteristic of R. The resolvent consists of a pair of locally free modules L, M with two multilinear maps between them; we can view L as Q/R, for Q the quintic ring, and M as S/R, where S is a sextic resolvent ring. As in Bhargava's treatment, any quintic ring has a resolvent ring, and for a maximal ring, the resolvent is unique. We hope that this work will enable the removal of the condition that the characteristic be different from 2 in Bhargava-Shankar-Wang's proof of Linnik's conjecture on the asymptotic distribution of discriminants of relative extensions. [ABSTRACT FROM AUTHOR]

This paper considers the problem of finding the resolvent of the sum of two maximal monotone operators. Such a problem arises frequently in practice, but it seems that computation of the solution of the problem is not necessarily easy. It is assumed that both the resolvents of two maximal monotone operators can be easily computed. This enables us to consider the case in which a solution to the problem cannot be computed easily. This paper introduces a new mapping, which satisfies the nonexpansivity property, from the individual resolvents of two maximal monotone operators and investigates some of its properties. In particular, we show that the mapping has a fixed point if and only if the problem has a solution. Then, using this mapping, we propose a splitting method for solving the problem in a real Hilbert space. In particular, we show that the sequences generated by the method converge strongly to the solution to the problem under certain assumptions. Convergence rate analysis of the methods is also provided to illustrate the method's efficiency. Finally, we apply the results to a class of optimization problems. [ABSTRACT FROM AUTHOR]

The paper is a continuation of Part I and contains several further results on generalized boundary triples, the corresponding Weyl functions, and applications of this technique to ordinary and partial differential operators. We establish a connection between Post's theory of boundary pairs of closed nonnegative forms on the one hand and the theory of generalized boundary triples of nonnegative symmetric operators on the other hand. Applications to the Laplacian operator on bounded domains with smooth, Lipschitz, and even rough boundary, as well as to mixed boundary value problem for the Laplacian are given. Other applications concern with the momentum, Schrödinger, and Dirac operators with local point interactions. These operators demonstrate natural occurrence of ES$ES$‐generalized boundary triples with domain invariant Weyl functions and essentially selfadjoint reference operators A0. [ABSTRACT FROM AUTHOR]

The present paper considers the operator pencil A (λ) = A 0 + A 1 λ , where A 0 , A 1 ≠ 0 are bounded linear mappings between complex Hilbert spaces and A 0 is neither one-to-one nor onto. Assuming that 0 is an isolated singularity of A (λ) and that the image of A 0 is closed, certain operators are defined recursively starting from A 0 and A 1 and they are shown to provide a characterization of the image and null space of the operators in the principal part of the resolvent and of the logarithmic residues of A (λ) at 0. The relations with the classical results based on ascent and descent in [10] are discussed. In the special case of A 0 being Fredholm of index 0, the present results characterize the rank of the operators in the principal part of the resolvent, the dimension of the subspaces that define the ascent and descent, the partial multiplicities, and the algebraic multiplicity of A (λ) at 0. [ABSTRACT FROM AUTHOR]

Motivated by the analysis of homogeneous operators and semigroups in [10] , in this paper we perform an spectral analysis of homogeneous operators and give simple characterization of generators of homogeneous semigroups and of homogeneous sectorial operators. We also analyse homogeneous perturbations of homogeneous operators and semigroups. We use these results to study some parabolic PDEs involving homogeneous operators including some fractional diffusion problems. We show a deep connection of these results with some Gagliardo–Nirenberg and Hardy type inequalities. [ABSTRACT FROM AUTHOR]

In this paper, we give a correct definition of the Laplace operator with delta-like potentials. Correctly solvable pointwise perturbation is investigated and formulas of resolvent are described. We study some properties of the resolvent. In particular, we prove Krein's formula for these resolvents. [ABSTRACT FROM AUTHOR]

Starting out from a new description of a class of parameter-dependent pseudodifferential operators with finite regularity number due to G. Grubb, we introduce a calculus of parameter-dependent, poly-homogeneous symbols whose homogeneous components have a particular type of point-singularity in the covariable-parameter space. Such symbols admit intrinsically a second kind of expansion which is closely related to the expansion in the Grubb–Seeley calculus and permits to recover the resolvent-trace expansion for elliptic pseudodifferential operators originally proved by Grubb–Seeley. Another application is the invertibility of parameter-dependent operators of Toeplitz type, i.e., operators acting in subspaces determined by zero-order pseudodifferential idempotents. [ABSTRACT FROM AUTHOR]

The aim of this paper is to give a precise proof of the completeness of Lamb modes of an elastic isotropic plate. This proof is relatively simple and short but relies on two powerful mathematical theorems. The first one is a theorem on elliptic systems with a parameter due to Agranovich and Vishik. The second one is a theorem due to Locker which gives a criterion to show the completeness of the set of generalized eigenvectors of a Hilbert–Schmidt discrete operator. [ABSTRACT FROM AUTHOR]

In this paper. we study the convergence analysis of Generalized Split Feasibility Problem (GSFP) in real Hilbert spaces. We introduce a new modified iterative algorithm which does not require any prior knowledge of operator norm for approximating the solution of GSFP and established the strong convergence of sequence generated by our iterative algorithm. The strong convergence result is proved without the prior knowledge of the operator norm. [ABSTRACT FROM AUTHOR]

We develop an algorithm that computes strongly continuous semigroups on infinite-dimensional Hilbert spaces with explicit error control. Given a generator A, a time t>0, an arbitrary initial vector u0 and an error tolerance ϵ>0, the algorithm computes exp(tA)u0 with error bounded by ϵ. The algorithm is based on a combination of a regularized functional calculus, suitable contour quadrature rules, and the adaptive computation of resolvents in infinite dimensions. As a particular case, we show that it is possible, even when only allowing pointwise evaluation of coefficients, to compute, with error control, semigroups on the unbounded domain L²(Rd) that are generated by partial differential operators with polynomially bounded coefficients of locally bounded total variation. For analytic semigroups (and more general Laplace transform inversion), we provide a quadrature rule whose error decreases like exp(-cN/log(N)) for N quadrature points, that remains stable as N→∞, and which is also suitable for infinite-dimensional operators. Numerical examples are given, including: Schrödinger and wave equations on the aperiodic Ammann--Beenker tiling, complex perturbed fractional diffusion equations on L²(R), and damped Euler--Bernoulli beam equations. [ABSTRACT FROM AUTHOR]

A large number of nonlinear and optimization problems can be reduced to altering point problems. This paper aims to introduce the parallel normal S-iteration technique and study its convergence rates for solving such problems in infinite-dimensional Hilbert spaces under practical assumptions. We place particular emphasis on the parallel splitting method for the sum of two maximal monotone operators and that can apply for solving a class of convex composite minimization problems. Moreover, we present applications of our iterative methods to some nonlinear problems, such as a system of variational inequalities and a system of inclusion problems. Finally, to demonstrate the applicability of the altering point technique, the performances of our proposed parallel normal S-iteration methods are presented through numerical experiments in signal recovery problems. [ABSTRACT FROM AUTHOR]

The motivation of our research is to establish a Laplace-domain theory that provides principles and methodology to analyze and synthesize systems with nonlinear dynamics. A semigroup of composition operators defined for nonlinear autonomous dynamical systems--the Koopman semigroup and its associated Koopman generator--plays a central role in this study. We introduce the resolvent of the Koopman generator, which we call the Koopman resolvent, and provide its spectral characterization for three types of nonlinear dynamics: ergodic evolution on an attractor, convergence to a stable equilibrium point, and convergence to a (quasi-)stable limit cycle. This shows that the Koopman resolvent provides the Laplace-domain representation of such nonlinear autonomous dynamics. A computational aspect of the Laplace-domain representation is also discussed with emphasis on nonstationary Koopman modes. [ABSTRACT FROM AUTHOR]

We consider equilibrium problems on a complete geodesic space with curvature bounded above by one and propose the notion of resolvents for this problem. We prove its well-definedness as a single-valued mapping whose domain is the whole space, and its geometric properties. [ABSTRACT FROM AUTHOR]

Using the resolvent operator, we develop an algorithm for computing smoothed approximations of spectral measures associated with self-adjoint operators. The algorithm can achieve arbitrarily high orders of convergence in terms of a smoothing parameter for computing spectral measures of general differential, integral, and lattice operators. Explicit pointwise and Lp-error bounds are derived in terms of the local regularity of the measure. We provide numerical examples, including a partial differential operator and a magnetic tight-binding model of graphene, and compute 1000 eigenvalues of a Dirac operator to near machine precision without spectral pollution. The algorithm is publicly available in SpecSolve, which is a software package written in MATLAB. [ABSTRACT FROM AUTHOR]

Some norm inequalities are presented for non- selfadjoint differential operators. First, the resolvent of the introduced non-selfadjoint differential operator using m-sectorial operators in the Hilbert space H = L 2 (Ω) is established. As application of this new result, the resolvent of the considered operator in the Hilbert space H ℓ = L 2 (Ω) ℓ is obtained utilizing some analytic techniques and diagonalizable way. [ABSTRACT FROM AUTHOR]

The adaptive Douglas-Rachford splitting algorithm iteratively applies the operator T = kΠ,QaQb ÷ (1 -- «n)Id to solve the inclusion problem zer(√4 + B). By taking a Yosida approximation standpoint, we express in canonical form QaQb -- (Id - (γ + λ) γ√4) o (Id -- (γ + λ) λB). We extend the domain of indices γ, λ to the entire real line, so that the adaptive algorithm is able to encompass a forward-backward splitting algorithm into one unified framework. Convergence results for both primal and dual problems are proved for different combinations of (strongly and weakly) monotone and comonotone operators. Under the "monotone + comonotone" assumption, we obtain a better rate bound for linear convergence than the classical Douglas-Rachford algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we discuss about monotone vector fields, which is a typical extension to the theory of convex functions, by exploiting the tangent space structure. This new approach to monotonicity in CAT (0) spaces stands in opposed to the monotonicity defined earlier in CAT (0) spaces by Khatibzadeh and Ranjbar [J. Aust. Math. Soc. 103(1), 70–90 (2017).] and Chaipunya and Kumam [Optimization 66(10), 1647–1665 (2017).]. In particular, this new concept extends the theory from both Hilbert spaces and Hadamard manifolds, while the known concept barely has any obvious relationship to the theory in Hadamard manifolds. We also study the corresponding resolvents and Yosida approximations of a given monotone vector field and derive many of their important properties. Finally, we prove a generation theorem by showing convergence of an exponential formula applied to resolvents of a monotone vector field. Our findings improve several known results in the literature including generation theorems of Jost [AMS/IP Stud. Adv. Math., vol. 8, pp. 1–47. Amer. Math. Soc., Providence, RI (1998)], Mayer [Comm. Anal. Geom. 6(2), 199–253 (1998).], Stojkovic [Adv. Calc. Var. 5(1), 77–126 (2012).], and Bačák [Convex analysis and optimization in Hadamard spaces, De Gruyter Series in Nonlinear Analysis and Applications, vol. 22. De Gruyter, Berlin (2014).] for proper, convex, lower semicontinuous functions in the context of complete CAT (0) spaces, and also by Iwamiya and Okochi [Nonlinear Anal. 54(2), 205–214 (2003).] for monotone vector fields in the context of Hadamard manifolds. [ABSTRACT FROM AUTHOR]

We consider a general scalar self-adjoint elliptic second order operator with general boundary conditions on an arbitrary metric graph containing a subgraph with edges of lengths proportional to a small parameter. We show that the resolvent of such operator is holomorphic in the small parameter and provide its representations by Taylor series. The coefficients of the series are found rather explicitly. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a second regularized trace formula on L2([0, π]; H) for a higher order self‐adjoint differential operator with unbounded operator‐valued coefficient, where H is a separable Hilbert space. [ABSTRACT FROM AUTHOR]

In this paper, we consider a differential inclusion governed by a set-valued nonexpansive mapping and study the asymptotic behavior (weak and strong convergence) of its solutions with various assumptions on this mapping. Then for a set-valued nonexpansive mapping, we define the corresponding resolvent (proximal) operator as a set-valued mapping and study some of its elementary properties. Subsequently, we apply the resolvent operator to state the implicit discretization of the differential inclusion and study the asymptotic behavior of its solutions which yields similar convergence results as in the continuous case. This provides an algorithm for approximating a fixed point of a set-valued nonexpansive mapping which extends the classical proximal point algorithm. An application to variational inequalities and a numerical comparison with another iterative method for approximating a fixed point of set-valued nonexpansive mappings are also presented. [ABSTRACT FROM AUTHOR]

In this paper, we study spectral decompositions with respect to a system of generalized eigenvectors of second-order differential operators on an interval whose resolvents possess convolution representations. We obtain the convolution representation of resolvents of second-order differential operators on an interval with integral boundary conditions. Then, using the convolution generated by the initial differential operator, we construct the Fourier transform. A connection between the convolution operation in the original functional space and the multiplication operation in the space of Fourier transforms is established. Finally, the problem on the convergence of spectral expansions generated by the original differential operator is studied. Examples of convolutions generated by operators are also presented. [ABSTRACT FROM AUTHOR]

We examine measures on a Banach space E that are invariant under shifts by arbitrary vectors of the space and are additive extensions of a set function defined on the family of bars with converging products of edge lengths that do not satisfy the σ-finiteness condition and, perhaps, the countable additivity condition. We introduce the Hilbert space ℋ of complex-valued functions of the space E of functions that are square integrable with respect to a shift-invariant measure. We analyze properties of semigroups of shift operators in the space ℋ and the corresponding generators and resolvents. We obtain a criterion of the strong continuity of such semigroups. We introduce and examine mathematical expectations of operators of shifts along random vectors by a one-parameter family of Gaussian measures that form a semigroup with respect to the convolution. We prove that the family of mathematical expectations is a one-parameter semigroup of linear self-adjoint contraction mappings of the space ℋ, find invariant subspaces of operators of this semigroup, and obtain conditions of its strong continuity. [ABSTRACT FROM AUTHOR]

We examine measures on a Banach space E that are invariant under shifts by arbitrary vectors of the space and are additive extensions of a set function defined on the family of bars with converging products of edge lengths that do not satisfy the σ-finiteness condition and, perhaps, the countable additivity condition. We introduce the Hilbert space ℋ of complex-valued functions of the space E of functions that are square integrable with respect to a shift-invariant measure. We analyze properties of semigroups of shift operators in the space ℋ and the corresponding generators and resolvents. We obtain a criterion of the strong continuity of such semigroups. We introduce and examine mathematical expectations of operators of shifts along random vectors by a one-parameter family of Gaussian measures that form a semigroup with respect to the convolution. We prove that the family of mathematical expectations is a one-parameter semigroup of linear self-adjoint contraction mappings of the space ℋ, find invariant subspaces of operators of this semigroup, and obtain conditions of its strong continuity. [ABSTRACT FROM AUTHOR]

Monotone operators and (firmly) nonexpansive mappings are fundamental objects in modern analysis and computational optimization. It was shown in 2012 that if finitely many firmly nonexpansive mappings have or "almost have" fixed points, then the same is true for compositions and convex combinations. More recently, sharp information about the minimal displacement vector of compositions and of convex combinations of firmly nonexpansive mappings was obtained in terms of the displacement vectors of the underlying operators. Using a new proof technique based on the Brezis–Haraux theorem and reflected resolvents, we extend these results from firmly nonexpansive to general averaged nonexpansive mappings. Various examples illustrate the tightness of our results. [ABSTRACT FROM AUTHOR]

The A-model for finite rank singular perturbations of class H-m-2∖H-m-1, m∈N, is considered from the perspective of boundary relations. Assuming further that the Hilbert spaces (Hn)n∈Z admit an orthogonal decomposition H-n⊕H+n, with the corresponding projections satisfying P±n+1⊆P±n, nontrivial extensions in the A-model are constructed for the symmetric restrictions in the subspaces. [ABSTRACT FROM AUTHOR]

Upper and lower bounds on the heat kernel on complete Riemannian manifolds were obtained in a series of pioneering works due to Cheng-Li-Yau, Cheeger-Yau and Li-Yau. However, these estimates do not give a complete picture of the heat kernel for all times and all pairs of points — in particular, there is a considerable gap between available upper and lower bounds at large distances and/or large times. Inspired by the work of Davies-Mandouvalos on H n + 1 , we study heat kernel bounds on Cartan-Hadamard manifolds that are asymptotically hyperbolic in the sense of Mazzeo-Melrose. Under the assumption of no eigenvalues and no resonance at the bottom of the continuous spectrum, we show that the heat kernel on such manifolds is comparable to the heat kernel on hyperbolic space of the same dimension (expressed as a function of time t and geodesic distance r), uniformly for all t ∈ (0 , ∞) and all r ∈ [ 0 , ∞). In particular our upper and lower bounds are uniformly comparable for all distances and all times. The corresponding statement for asymptotically Euclidean spaces is not known to hold, and as we argue in the last section, it is very unlikely to be true in that geometry. As an application, we show boundedness on Lp of the Riesz transform ∇ (Δ − n 2 / 4 + λ 2) − 1 / 2 , for λ ∈ (0 , n / 2 ] , on such manifolds, for p satisfying | p − 1 − 2 − 1 | < λ / n. For λ = n / 2 (the standard Riesz transform ∇ Δ − 1 / 2 ), this was previously shown by Lohoué in a more general setting. [ABSTRACT FROM AUTHOR]

Kato's theory, which relies on the analytical properties of the resolvent of a differential operator, is used in this paper to obtain a closed‐form perturbation solution of an electromechanical free‐vibration problem of coupled piezoelectricity. Making use of the fact that piezoelectric coupling is not too strong, the natural frequencies and the mode shapes of the through‐thickness vibration of an infinite piezoelectric plate are calculated using a perturbation method. Explicit perturbation formulae have been derived up to the third‐order terms for the natural frequencies, whereas the closed‐form perturbation expressions for the mode shapes have been obtained up to the first‐order terms through the use of a projection operator. Numerical results are discussed for a plate with fixed‐free and free‐free faces. Since exact solutions exist for the problem in hand, the validity of the proposed method can be proved. The method differs from the perturbation approach usually used in quantum mechanics, since it does not rely on the expansion of the perturbation solution in series of the unperturbed eigenmodes; since the solutions are obtained in closed form, no issues of the convergence of infinite series arise. The perturbation approach used allows to obtain an approximate solution of a coupled problem by solving only the corresponding uncoupled elastic problem. The paper is a good illustration of the application of Kato's analytical method to solving a physical problem, which may be of interest for other vibration problems, too. [ABSTRACT FROM AUTHOR]

With a closed symmetric operator A in a Hilbert space H a triple Π={H,Γ0,Γ1} of a Hilbert space H and two abstract trace operators Γ0 and Γ1 from A∗ to H is called a generalized boundary triple for A∗ if an abstract analogue of the second Green's formula holds. Various classes of generalized boundary triples are introduced and corresponding Weyl functions M(·) are investigated. The most important ones for applications are specific classes of boundary triples for which Green's second identity admits a certain maximality property which guarantees that the corresponding Weyl functions are Nevanlinna functions on H, i.e. M(·)∈R(H), or at least they belong to the class R∼(H) of Nevanlinna families on H. The boundary condition Γ0f=0 determines a reference operator A0(=kerΓ0). The case where A0 is selfadjoint implies a relatively simple analysis, as the joint domain of the trace mappings Γ0 and Γ1 admits a von Neumann type decomposition via A0 and the defect subspaces of A. The case where A0 is only essentially selfadjoint is more involved, but appears to be of great importance, for instance, in applications to boundary value problems e.g. in PDE setting or when modeling differential operators with point interactions. Various classes of generalized boundary triples will be characterized in purely analytic terms via the Weyl function M(·) and close interconnections between different classes of boundary triples and the corresponding transformed/renormalized Weyl functions are investigated. These characterizations involve solving direct and inverse problems for specific classes of operator functions M(·). Most involved ones concern operator functions M(·)∈R(H) for which τM(λ)(f,g)=(2iImλ)−1[(M(λ)f,g)−(f,M(λ)g)],f,g∈domM(λ),defines a closable nonnegative form on H. It turns out that closability of τM(λ)(f,g) does not depend on λ∈C± and, moreover, that the closure then is a form domain invariant holomorphic function on C± while τM(λ)(f,g) itself need not be domain invariant. In this study we also derive several additional new results, for instance, Kreĭn‐type resolvent formulas are extended to the most general setting of unitary and isometric boundary triples appearing in the present work. In part II of the present work all the main results are shown to have applications in the study of ordinary and partial differential operators. [ABSTRACT FROM AUTHOR]

By using our own approach, we study the strong convergence of an inexact proximal point algorithm with possible unbounded errors for a maximal monotone operator in a Banach space. We give a necessary and sufficient condition for the zero set of the operator to be nonempty and show that, in this case, this iterative sequence converges strongly to a zero of the operator. We present also some applications of our results to equilibrium problems and optimization. Our proximal point algorithm contains, as a special case, the one considered in Hilbert space by Djafari Rouhani and Moradi in (J Optim Theory Appl 172:222–235, 2017) and solves the open problem of extending it to a Banach space, which was stated in that paper and in Djafari Rouhani and Moradi in (J Optim Theory Appl 181:864–882, 2019). Since the nonexpansiveness of the resolvent operator, which holds in Hilbert space, is not valid anymore in Banach space, our results require new methods of proofs, and significantly improve upon the previous results, both in theory and in applications. [ABSTRACT FROM AUTHOR]