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The aim of this paper is to study the completeness and basicity problems for selfadjoint operators of the class K(H) in almost Krein spaces and prove criteria for the basicity and completeness of root vectors of linear pencils. [ABSTRACT FROM AUTHOR]

Throughout this chapter, the field $$ \mathbb{K} $$ will always be the real field ℝ; we consider a real Banach space U, an open interval ℭ ⊂ ∝, a neighborhood $$ \mathcal{U} $$ of 0 ∈ U, an integer number r ≥ 0, a family $$ \mathfrak{L} $$∈Cr(Ω,$$ \mathcal{L} $$(U)), and a nonlinear map $$ \mathfrak{N} $$∈ C(Ω × $$ \mathcal{U} $$, U) satisfying the following conditions: (AL)$$ \mathfrak{L} $$(λ) ™ IU∈ K(U) for every λ ∈ Ω, i.e., $$ \mathfrak{L} $$(λ) is a compact perturbation of the identity map. (AN)$$ \mathfrak{N} $$ is compact, i.e., the image by $$ \mathfrak{N} $$ of any bounded set of Ω × $$ \mathcal{U} $$ is relatively compact in U. Also, for every compact K ⊂ Ω, $$ \mathop {\lim }\limits_{u \to 0} \mathop {\sup }\limits_{\lambda \in K} \frac{{\left\ {\mathfrak{N}\left( {\lambda ,u} \right)} \right\}} {{\left\ u \right\}} = 0. $$. From now on, we consider the operator $$ \mathfrak{F} \in \mathcal{C}\left( {\Omega \times \mathcal{U},U} \right) $$ defined as 12.1$$ \mathfrak{F}\left( {\lambda ,u} \right): = \mathfrak{L}\left( \lambda \right)u + \mathfrak{N}\left( {\lambda ,u} \right), $$ and the associated equation 12.2$$ \begin{array}{*{20}c} {\mathfrak{F}\left( {\lambda ,u} \right) = 0,} & {\left( {\lambda ,u} \right) \in \Omega } \\ \end{array} \times \mathcal{U}. $$ By Assumptions (AL) and (AN), it is apparent that $$ \begin{array}{*{20}c} {\mathfrak{F}\left( {\lambda ,u} \right) = 0,} & {D_u \mathfrak{F}\left( {\lambda ,u} \right) = \mathfrak{L}\left( \lambda \right),} & \lambda \\ \end{array} \in \Omega , $$ and, hence, (12.2) can be thought of as a nonlinear perturbation around (λ, 0) of the linear equation 12.3$$ \begin{array}{*{20}c} {\mathfrak{L}\left( \lambda \right)u = 0,} & {\lambda \in \Omega ,} & u \\ \end{array} \in U. $$ Equation (12.2) can be expressed as a fixed-point equation for a compact operator. Indeed, $$ \mathfrak{F}\left( {\lambda ,u} \right) $$ = 0 if and only if $$ u = \left[ {I_U - \mathfrak{L}\left( \lambda \right)} \right]u - \mathfrak{N}\left( {\lambda ,u} \right). $$. [ABSTRACT FROM AUTHOR]

This chapter studies the polynomial families of the form 10.1$$ \begin{array}{*{20}c} {\mathfrak{L}\left( \lambda \right) = \sum\limits_{j = 0}^\ell {A_j \lambda ^j } ,} & {\lambda \in \mathbb{C}} \\ \end{array} , $$ where ℓ ∈ ℕ and $$ \begin{array}{*{20}c} {A_0 , \ldots ,A_\ell \in \mathcal{M}_N \left( \mathbb{C} \right),} & {A_\ell \ne 0,} \\ \end{array} $$ for some N ∈ ℕ*. These families are called matrix polynomials in most of the available literature. More precisely, the family $$ \mathfrak{L} $$ defined in (10.1) is said to be a matrix polynomial of order N and degree ℓ. The main goal of this chapter is to obtain a spectral theorem for matrix polynomials, respecting the spirit of the Jordan Theorem 1.2.1. [ABSTRACT FROM AUTHOR]

This chapter exposes briefly some further developments of the theory of multiplicity whose treatment lies outside the general scope of this book. Consequently, it possesses an expository and bibliographic character. [ABSTRACT FROM AUTHOR]

The stability results of Section 8.4 can be regarded as infinite-dimensional versions of the classic Rouché theorem. A closely related topic in complex function theory is the so-called argument principle, otherwise known as the logarithmic residue theorem, which has been established by Theorem 3.4.1 (for classical families) and Corollary 6.5.2 in a finite-dimensional setting. [ABSTRACT FROM AUTHOR]

This chapter focuses attention on the analytic operator families. After studying some universal spectral properties of these families, this chapter deals with the classical families $$ \mathfrak{L}^A $$ of the form 8.1$$ \begin{array}{*{20}c} {\mathfrak{L}^A \left( \lambda \right): = \lambda I_U - A,} & {\lambda \in \mathbb{K}} \\ \end{array} , $$ for a given A ∈$$ A \in \mathcal{L} $$(U), in order to show that, in this particular case, the classic concepts of algebraic ascent and multiplicity equal the generalized concepts introduced in the previous four chapters. Consequently, the algebraic multiplicity analyzed in this book, from a series of different perspectives, is indeed a generalization of the classic concept of algebraic multiplicity. [ABSTRACT FROM AUTHOR]

This chapter studies the algebraic multiplicity according to the theory of Jordan chains. The concept of Jordan chain extends the classic concept of chain of generalized eigenvectors, already studied in Section 1.3. It will provide us with a further approach to the algebraic multiplicities χ and µ introduced and analyzed in Chapters 4 and 5, respectively, whose axiomatization has already been accomplished through the uniqueness theorems included in Chapter 6. [ABSTRACT FROM AUTHOR]

Throughout this chapter, given $$ \mathbb{K} \in \left\{ {\mathbb{R},\mathbb{C}} \right\} $$, two non-zero $$ \mathbb{K} $$-Banach spaces U, V, and $$ \lambda _0 \in \mathbb{K} $$, we denote by $$ \mathcal{S}_{\lambda _0 }^\infty \left( {U,V} \right) $$ the set of all families $$ \mathfrak{L} $$ of class C∞ in a neighborhood of λ0 with values in $$ \mathcal{L}\left( {U,V} \right) $$ such that $$ \mathfrak{L}_0 : = \mathfrak{L}\left( {\lambda _0 } \right) \in Fred_0 \left( {U,V} \right); $$ the neighborhood may depend on $$ \mathfrak{L} $$. We also set $$ \mathcal{S}_{\lambda _0 }^\infty \left( U \right): = \mathcal{S}_{\lambda _0 }^\infty \left( {U,U} \right). $$ Clearly, $$ \mathfrak{L}\mathfrak{M} \in \mathcal{S}_{\lambda _0 }^\infty \left( {U,V} \right) $$ if $$ \mathfrak{L}^{ - 1} \in \mathcal{S}_{\lambda _0 }^\infty \left( {W,V} \right) $$ and $$ \mathfrak{M} \in \mathcal{S}_{\lambda _0 }^\infty \left( {U,W} \right) $$, where W is another $$ \mathbb{K} $$-Banach space. Moreover, $$ \mathfrak{L}^{ - 1} \in \mathcal{S}_{\lambda _0 }^\infty \left( {V,U} \right) $$ whenever $$ \mathfrak{L} \in \mathcal{S}_{\lambda _0 }^\infty \left( {U,V} \right) $$ and $$ \mathfrak{L}_0 \in Iso\left( {U,V} \right) $$ Iso(U, V). [ABSTRACT FROM AUTHOR]

This chapter describes an equivalent approach to the concept of multiplicity $$ \chi \left[ {\mathfrak{L};\lambda _0 } \right] $$ introduced in Chapter 4; in this occasion by means of an appropriate polynomial factorization of $$ \mathfrak{L} $$ at λ0. However, at first glance these approaches are seemingly completely different. [ABSTRACT FROM AUTHOR]

Throughout this chapter we will consider $$ \mathbb{K} \in \left\{ {\mathbb{R},\mathbb{C}} \right\} $$, two $$ \mathbb{K} $$-Banach spaces U and V, an open subset $$ \Omega \subset \mathbb{K} $$, a point λ0 ∈ Ω, and a family $$ \mathfrak{L} \in \mathcal{C}^r \left( {\Omega ,\mathcal{L}\left( {U,V} \right)} \right), $$ for some r ∈ ℕ ∪ {∞}, such that $$ \mathfrak{L}_0 : = \mathfrak{L}\left( {\lambda _0 } \right) \in Fred_0 \left( {U,V} \right). $$ When λ0 ∈ Eig$$ \left( \mathfrak{L} \right) $$, the point λ0 is said to be an algebraic eigenvalue of $$ \mathfrak{L} $$ if there exist δ, C > 0 and m ≥ 1 such that, for each 0 < λ − λ0 < δ, the operator $$ \mathfrak{L}\left( \lambda \right) $$ is an isomorphism and $$ \left\ {\mathfrak{L}\left( \lambda \right)^{ - 1} } \right\ \leqslant \frac{C} {{\left {\lambda - \lambda _0 } \right^m }}. $$ The main goal of this chapter is to introduce the concept of algebraic multiplicity of $$ \mathfrak{L} $$ at any algebraic eigenvalue λ0. This algebraic multiplicity will be denoted by $$ \chi \left[ {\mathfrak{L};\lambda _0 } \right] $$, and will be defined through the auxiliary concept of transversal eigenvalue. Such concept will be motivated in Section 4.1 and will be formally defined in Section 4.2. Essentially, λ0 is a transversal eigenvalue of $$ \mathfrak{L} $$ when it is an algebraic eigenvalue for which the perturbed eigenvalues $$ a\left( \lambda \right) \in \sigma \left( {\mathfrak{L}\left( \lambda \right)} \right) $$ from $$ 0 \in \sigma \left( {\mathfrak{L}_0 } \right) $$, as λ moves from λ0, can be determined through standard perturbation techniques; these perturbed eigenvalues a(λ) are those satisfying a(λ0) = 0. This feature will be clarified in Sections 4.1 and 4.4, where we study the behavior of the eigenvalue a(λ) and its associated eigenvector in the special case when 0 is a simple eigenvalue of $$ \mathfrak{L}_0 $$. In such a case, the multiplicity of $$ \mathfrak{L} $$ at λ0 equals the order of the function a at λ0. [ABSTRACT FROM AUTHOR]

This chapter considers $$ A \in \mathcal{M}_N \left( \mathbb{C} \right) $$ and uses the Dunford integral formula to construct the spectral projections of ℂN onto N[(A − λI)ν(λ)] for every λ ∈ σ(A). It also shows that, for each λ ∈ σ(A), the algebraic ascent ν(λ) equals the order of λ as a pole of the associated resolvent operator 3.1$$ \begin{array}{*{20}c} {\mathcal{R}\left( {z;A} \right): = \left( {zI - A} \right)^{ - 1} ,} & {z \in \rho \left( A \right): = \mathbb{C}\backslash \sigma \left( A \right)} \\ \end{array} . $$ Precisely, this chapter is structured as follows. Section 3.1 gives a universal estimate for the norm of the inverse of a matrix in terms of its determinant and its norm. From this estimate it will become apparent that the eigenvalues of A are poles of the resolvent operator (3.1). The necessary analysis to show this feature will be carried out in Section 3.3. Section 3.2 gives a result on Laurent series valid for vector-valued holomorphic functions. Finally, Section 3.4 constructs the spectral projections associated with the direct sum decomposition (1.6), whose validity was established by the Jordan Theorem 1.2.1. [ABSTRACT FROM AUTHOR]

In this chapter we prove the Jordan theorem, a pivotal result in mathematics, which establishes that, for every $$ A \in \mathcal{M}_N \left( \mathbb{C} \right) $$, the space ℂN decomposes as the direct sum of the ascent generalized eigenspaces associated with each of the eigenvalues of A. Then, by choosing an appropriate basis in each of the ascent generalized eigenspaces, the Jordan canonical form of A is constructed. These bases are chosen in order to attain a similar matrix to A with a maximum number of zeros. [ABSTRACT FROM AUTHOR]

In Section 1.2 we defined P(A) when P is a polynomial and $$ A \in \mathcal{M}_N \left( \mathbb{K} \right) $$ with $$ \mathbb{K} \in \left\{ {\mathbb{R},\mathbb{C}} \right\} $$. More generally, one of the main goals of operator calculus consists in giving an appropriate definition of f(A) when $$ f:\mathbb{K} \to \mathbb{K} $$ is an arbitrary function, as well as in studying the most important analytical properties of f(A). This chapter covers these issues for the special, but important, case when f is a certain holomorphic function and $$ A \in \mathcal{M}_N \left( \mathbb{C} \right) $$. [ABSTRACT FROM AUTHOR]

We discuss the definition of a rank one singular perturbation of a non-self-adjoint operator L in Hilbert space H. Provided that the operator L is a non-self-adjoint perturbation of a self-adjoint operator A and that the spectrum of the operator L is absolutely continuous we are able to establish a concise resolvent formula for the singular perturbations of the class considered and to establish a model representation of it in the dilation space associated with the operator L. [ABSTRACT FROM AUTHOR]

We study operator-valued functions of weighted Schur classes over multiply-connected domains. There is a correspondence between functions of weighted Schur classes and so-called "conservative curved" systems introduced in the paper. In the unit disk case the fundamental relationship between invariant subspaces of the main operator of a conservative system and factorizations of the corresponding Schur class function (characteristic function) is well known. We extend this connection to weighted Schur classes. With this aim we develop new notions and constructions and make suitable changes in standard theory. [ABSTRACT FROM AUTHOR]

We consider self-adjoint unbounded Jacobi matrices with diagonal qn = n and weights λn = cnn, where cn is a 2-periodical sequence of real numbers. The parameter space is decomposed into several separate regions, where the spectrum is either purely absolutely continuous or discrete. This constitutes an example of the spectral phase transition of the first order. We study the lines where the spectral phase transition occurs, obtaining the following main result: either the interval (−∞; 1/2) or the interval (1/2; +∞) is covered by the absolutely continuous spectrum, the remainder of the spectrum being pure point. The proof is based on finding asymptotics of generalized eigenvectors via the Birkhoff-Adams Theorem. We also consider the degenerate case, which constitutes yet another example of the spectral phase transition. [ABSTRACT FROM AUTHOR]

Uniform and smooth asymptotics for the solutions of a parametric system of difference equations are obtained. These results are the uniform and smooth generalizations of the Benzaid-Lutz theorem (a Levinson type theorem for discrete linear systems) and are used to develop a technique for proving absence of accumulation points in the pure point spectrum of Jacobi matrices. The technique is illustrated by proving discreteness of the spectrum for a class of unbounded Jacobi operators. [ABSTRACT FROM AUTHOR]

Anomalies are known to appear in the perturbation theory for the one-dimensional Anderson model. A systematic approach to anomalies at critical points of products of random matrices is developed, classifying and analysing their possible types. The associated invariant measure is calculated formally. For an anomaly of so-called second degree, it is given by the ground-state of a certain Fokker-Planck equation on the unit circle. The Lyapunov exponent is calculated to lowest order in perturbation theory with rigorous control of the error terms. [ABSTRACT FROM AUTHOR]

This paper offers the functional model of a class of non-selfadjoint extensions of a symmetric operator with equal deficiency indices. The explicit form of dilation of a dissipative extension is offered and the Sz.-Nagy-Foiaş model as developed by B. Pavlov is constructed. A variant of functional model for a non-selfadjoint non-dissipative extension is formulated. We illustrate the theory by two examples: singular perturbations of the Laplace operator in L2(ℝ3) by a finite number of point interactions, and the Schrödinger operator on the half-axis (0, ∞) in the Weyl limit circle case at infinity. [ABSTRACT FROM AUTHOR]

In this paper we study the problem of unique reconstruction of the quantum graphs. The idea is based on the trace formula which establishes the relation between the spectrum of Laplace operator and the set of periodic orbits, the number of edges and the total length of the graph. We analyse conditions under which is it possible to reconstruct simple graphs containing edges with rationally dependent lengths. [ABSTRACT FROM AUTHOR]

Plasma waves in a slot-diod with governing electrodes are described by the linearized hydrodynamic equations. Separation of variables in the corresponding scattering problem is generally impossible. Under natural physical assumption we reduce the problem to the second order differential equation on the slot with an operator weight, defined by the Dirichlet-to-Neumann map of the three-dimensional Laplacian on the complement of the electrodes and the slot. The reduction is based on a formula for the Poisson map for the exterior Laplace Dirichlet problem on the complement of a few standard bodies in terms of the Poisson maps on the complement of each standard body. [ABSTRACT FROM AUTHOR]

We investigate trace formulas for Jacobi operators which are trace class perturbations of quasi-periodic finite-gap operators using Krein's spectral shift theory. In particular we establish the conserved quantities for the solutions of the Toda hierarchy in this class. [ABSTRACT FROM AUTHOR]

We consider Schrödinger operators H = −Δ + V in L2(Ω) where the domain Ω ⊂ ℝ+d+1 and the potential V = V (x, y) are periodic with respect to the variable x ∈ ℝd. We assume that Ω is unbounded with respect to the variable y ∈ ℝ and that V decays with respect to this variable. V may contain a singular term supported on the boundary. We develop a scattering theory for H and present an approach to prove absence of singular continuous spectrum. Moreover, we show that certain repulsivity conditions on the potential and the boundary of Ω exclude the existence of surface states. In this case, the spectrum of H is purely absolutely continuous and the scattering is complete. [ABSTRACT FROM AUTHOR]

We use a caricature model of a system consisting of a quantum wire and a finite number of quantum dots, to discuss relation between the Zeno dynamics and the stable one which governs time evolution of the dot states in the absence of the wire. We analyze the weak coupling case and argue that the two time evolutions can differ significantly only at times comparable with the lifetime of the unstable system undisturbed by perpetual measurement. [ABSTRACT FROM AUTHOR]

In this paper we present some results obtained recently, partly in collaboration with Didier Robert, about "quantum revivals" and "quantum fidelity", mainly in the semiclassical framework. We also describe the exact properties of the quantum fidelity (also called Loschmidt echo) for the case of explicit quadratic plus inverse quadratic time-periodic Hamiltonians and establish that the quantum fidelity equals one for exactly the times where the classical fidelity is maximal. [ABSTRACT FROM AUTHOR]

Finiteness criteria are established for the point spectrum of the perturbed Dirac operator. The results are obtained by applying the direct methods of the perturbation theory of linear operators. The particular case of the Hamiltonian of a Dirac particle in an electromagnetic field is also considered. [ABSTRACT FROM AUTHOR]

New presheaves of hyperfunction spaces with the growth estimates with respect to x → ∞ and y → 0 in a cone Γ are introduced. Then it is shown that the Laplace transform is a bijective mapping of the space of tempered ultradistributions on Rn of non-quasianalytic class onto the corresponding hyperfunction space of sections over Dn, the compactification of Rn. Microlocalization of tempered ultradistributions at (x0∞, ξ0) is introduced as well as a new microlocalization within some classes of hyperfunctions. [ABSTRACT FROM AUTHOR]

Let M(ω)p,q be the modulation space with parameters p, q and weight function ω. We prove that if p, q, p1, p2, q1, q2 ∈ [1,∞], ω1, ω2, ω and h1, h2 are appropriate, and a ∈ M(ω)p,q, then the Toeplitz operator $$ Tp_{h_1 ,h_2 } (a):M_{(\omega _1 )}^{p_1 ,q_1 } \to M_{(\omega _2 )}^{p_2 ,q_2 } $$ is continuous. If in addition p1 = p2 = q1 = q2 = 2, then we present sufficient conditions on p, q, h1 and h2 in order for $$ Tp_{h_1 ,h_2 } (a) $$ should be a Schatten-von Neumann operator of certain degree. [ABSTRACT FROM AUTHOR]

We present new results concerning pseudo-differential operators in the function spaces Sμμ(ℝn) of Gelfand and Shilov. In particular we discuss Sνμ(ℝn)-regularity of solutions to SG-elliptic pseudo-differential equations, allowing lower order semilinear perturbations. The results apply to SG-elliptic partial differential equations with polynomial coefficients. We also study the action of Weyl operators and localization operators on Sνμ(ℝn). [ABSTRACT FROM AUTHOR]

We provide examples of the product of two localization operators. As a special case, we study the composition of Gabor multipliers. The results highlight the instability of this product and underline the necessity of expressing it in terms of asymptotic expansions. [ABSTRACT FROM AUTHOR]

A large class of time-varying filters can be described via pseudo-differential operators belonging to the Hörmander class OPS00, 0. The questions whether and how an input signal can be reconstructed from a known output lead to the problems of invertibility of pseudo-differential operators in that class and of (at least, numerical) solution of pseudo-differential equations. We are going to derive effective conditions for the invertibility for pseudo-differential operators with globally slowly varying symbols as well as for causal pseudo-differential operators, and we study the stability of the finite sections method with respect to time and frequency for these operators. [ABSTRACT FROM AUTHOR]

Signals in real applications are typically finite in duration, dynamic and non-stationary processes with frequency characteristics varying over time. This often requires techniques capable of locally analyzing and processing signals. An integral transform known as the Stockwell transform is a combination of the classic Gabor transform and the current and versatile wavelet transform. It allows more accurate detection of subtle changes and easy interpretation in the time-frequency domain. In this paper, we study the mathematical underpinnings of the Stockwell transform. We look at the Stockwell transform as a stack of simple pseudo-differential operators parameterized by frequencies and give a complete description of the Stockwell spectra. [ABSTRACT FROM AUTHOR]

Motivated by problems in signal analysis, we define a class of time-frequency representations which is based on the short-time Fourier transform and depends on two fixed windows. We show that this class can be viewed as a link between the classical Rihaczek representation and the spectrogram. Correspondingly we formulate for this class a suitable general form of the uncertainty principle which have, as limit case, the uncertainty principles for the Rihaczek representation and for the spectrogram. We finally consider the questions of marginal distributions. We compute them in terms of convolutions with the windows and prove simple conditions for which average and standard deviation of the distributions in our class coincide with that of their marginals. [ABSTRACT FROM AUTHOR]

Using the boundedness of the maximal singular integral operator related to the Carleson-Hunt theorem we prove the boundedness and study the compactness of pseudo-differential operators a(x,D) with bounded measurable V (ℝR)-valued symbols a(x, ·) on the Lebesgue spaces Lp(ℝ) with 1 < p < δ, where V (ℝ) is the Banach algebra of all functions of bounded total variation on R. Replacement of absolutely continuous functions of bounded total variation by arbitrary functions of bounded total variation allows us to study pseudo-differential operators with symbols admitting discontinuities of the first kind with respect to the spatial and dual variables. Appearance of discontinuous symbols leads to non-commutative algebras of Fredholm symbols. Three different Banach algebras of pseudo-differential operators with discontinuous symbols acting on the spaces Lp(ℝ) are studied. We construct Fredholm symbol calculi for these algebras and establish Fredholm criteria for the operators in these algebras in terms of their Fredholm symbols. For the operators in the first algebra we also obtain an index formula. An application to the Haseman boundary value problem is given. [ABSTRACT FROM AUTHOR]

Let M(ω)p,q be the modulation space with parameters p, q and weight function ω. We prove that if t ∈ R, p, pj, q, qj ∈ [1, ∞], ω1, ω2 and ω are appropriate, and a ∈ M(ω)p,q, then the pseudo-differential operator at(x,D) is continuous from M(ω)p1,q1 to M(ω)p2,q2. If in addition pj = qj = 2, then we establish necessary and sufficient conditions on p and q in order to at(x,D) should be a Schatten-von Neumann operator of certain order. [ABSTRACT FROM AUTHOR]

In this paper a result of continuity for pseudo-differential operators with non-regular symbols on spaces of quasi-homogeneous type is given. More precisely, the symbols a(x, ξ) take their values in a quasi-homogeneous Besov space with respect to the x variable; moreover a finite number of derivatives with respect to the second variable satisfies, in Besov norm, decay estimates of quasi-homogeneous type. [ABSTRACT FROM AUTHOR]

For the differential operator P of order m and the inhomogeneous data f ∈ S′ on Rn, we say that the Lp-extension of the solution holds if u ∈ Lp, m ≤ n(1 − 1/p), and Pu = f on Rn / 0 imply Pu = f on Rn. In this article, we discuss which kind of inhomogeneous terms f ∈ S′ admit the Lp-extension of the solution. In previous works, this problem was studied by using classical Bochner's method ([1]) or a new method developed by the author and Uchida ([5], [4]). We consider inhomogeneous terms which are not covered by these results. [ABSTRACT FROM AUTHOR]

In this paper we study the local solvability in Gevrey classes for degenerate parabolic operators of order ≥ 2. We assume that the lower order term vanishes at a suitably smaller rate with respect to the principal part; we then analyze its influence on the behavior of the operator, proving local solvability in Gevrey spaces Gs for small s, and local nonsolvability in Gs for large s. [ABSTRACT FROM AUTHOR]

We study the exponential decay of solutions to differential equations of hypoelliptic type (see Definition 2.1). In particular we find sufficient conditions on the differential operator A in order for estimates of the kind $$ e^{\varepsilon \left\langle x \right\rangle ^r } Au \in V \Rightarrow e^{\varepsilon \left\langle x \right\rangle ^r } u \in V $$ to hold, for several types of functions and distribution spaces V. For example, V may be . [ABSTRACT FROM AUTHOR]

We construct the wave kernels of the non-isotropic twisted Laplacian by means of its heat kernel. We then express the wave kernels at the origin in terms of complex Fourier integrals and we exploit the connections of the phase functions with the complex integrals. [ABSTRACT FROM AUTHOR]

Basic properties of Fourier integral operators on the torus $$ \mathbb{T}^n = (\mathbb{R}/2\pi \mathbb{Z})^n $$ are studied by using the global representations by Fourier series instead of local representations. The results can be applied in studying hyperbolic partial differential equations. [ABSTRACT FROM AUTHOR]

The Plancherel formula and the inversion formula for Weyl transforms on compact and Hausdorff groups are given. A formula expressing the relationships of the wavelet constant, the degree of the irreducible and unitary representation and the volume of an arbitrary compact and Hausdorff group is derived. The role of the Weyl transforms in the derivation of the formulas for the heat kernels of Laplacians on compact Lie groups is explicated. The Green functions and the Riemann zeta functions of Laplacians on compact Lie groups are constructed using the corresponding heat kernels. [ABSTRACT FROM AUTHOR]

The method of construction of the fundamental solution for heat equations using pseudo-differential operators with parameter time variable is discussed, which is applicable to calculate traces of operators. This gives extensions of both the Gauss-Bonnet-Chern Theorem and the Riemann-Roch Theorem. [ABSTRACT FROM AUTHOR]

We present an overview of some of our recent results on the existence of rays of minimal growth for elliptic cone operators and two new results concerning the necessity of certain conditions for the existence of such rays. [ABSTRACT FROM AUTHOR]

We investigate operators on manifolds with edges from the point of view of the symbolic calculus induced by the singularities. We discuss new aspects of the quantization of edge-degenerate symbols which lead to continuous operators in weighted edge spaces. [ABSTRACT FROM AUTHOR]

For a class of pairs of entire matrix functions the null space of the natural analogue of the classical resultant matrix is described in terms of the common Jordan chains of the defining entire matrix functions. The main theorem is applied to two inverse problems. The first concerns convolution integral operators on a finite interval with matrix valued kernel functions and complements earlier results of [6]. The second is the inverse problem for matrix-valued continuous analogues of Szegő orthogonal polynomials. [ABSTRACT FROM AUTHOR]

We study the behavior of the sequence of minimal vectors corresponding to certain classes of operators on L2 spaces, including weighted composition operators such as those induced by Möbius transformations. In conjunction with criteria for quasinilpotence, the convergence of sequences associated with the minimal vectors leads to the construction of hyperinvariant subspaces. [ABSTRACT FROM AUTHOR]

The paper is concerned with a sequence of constants which appear in several problems. These problems include the minimal eigenvalue of certain positive definite Toeplitz matrices, the minimal eigenvalue of some higher-order ordinary differential operators, the norm of the Green kernels of these operators, the best constant in a Wirtinger-Sobolev inequality, and the conditioning of a special least squares problem. The main result of the paper gives the asymptotics of this sequence. [ABSTRACT FROM AUTHOR]

The central result of this paper is a method of characterizing those commuting tuples of operators that have a unitary dilation, in terms of the existence of a positive map with certain properties. Although this positivity condition is not necessarily easy to check given a concrete example, it can be used to find practical tests in some circumstances. As an application, we extend a dilation theorem of Sz.-Nagy and Foiaş concerning regular dilations to a more general setting [ABSTRACT FROM AUTHOR]

Notions of weak and uniformly weak mixing (to zero) are defined for bounded sequences in arbitrary Banach spaces. Uniformly weak mixing for vector sequences is characterized by mean ergodic convergence properties. This characterization turns out to be useful in the study of multiple recurrence, where mixing properties of vector sequences, which are not orbits of linear operators, are investigated. For bounded sequences, which satisfy a certain domination condition, it is shown that weak mixing to zero is equivalent with uniformly weak mixing to zero. [ABSTRACT FROM AUTHOR]