Kato Type Research Articles

Polaroid operators satisfying Weyl’s theorem

A Banach space operator T is polaroid and satisfies Weyl’s theorem if and only if T is Kato type at points λ ∈ iso σ( T) and has SVEP at points λ not in the Weyl spectrum of T. For such operators T, f( T) satisfies Weyl’s theorem for every non-constant function f analytic on a neighborhood of σ( T) if and only if f( T ∗) satisfies Weyl’s theorem.

Tensor products, multiplications and Weyl’s theorem

Tensor productsZ=T 1⊗T 2 and multiplicationsZ=L T 1 R T 2 do not inherit Weyl’s theorem from Weyl’s theorem forT 1 andT 2. Also, Weyl’s theorem does not transfer fromZ toZ*. We prove that ifT i,i=1, 2, has SVEP (=the single-valued extension property) at points in the complement of the Weyl spectrumσ w(Ti) ofT i, and if the operatorsT i are Kato type at the isolated points ofσ(Ti), thenZ andZ* satisfy Weyl’s theorem.

Kato type operators and Weyl's theorem

A Banach space operator T satisfies Weyl's theorem if and only if T or T ∗ has SVEP at all complex numbers λ in the complement of the Weyl spectrum of T and T is Kato type at all λ which are isolated eigenvalues of T of finite algebraic multiplicity. If T ∗ (respectively, T) has SVEP and T is Kato type at all λ which are isolated eigenvalues of T of finite algebraic multiplicity (respectively, T is Kato type at all λ ∈ iso σ ( T ) ), then T satisfies a-Weyl's theorem (respectively, T ∗ satisfies a-Weyl's theorem).

Feynman-Kac propagators and viscosity solutions

The aim of this paper is to study viscosity solutions to the following terminal value problem on [0, t] × E: $$ \left\{ {\begin{array}{*{20}l} {\frac{{\partial u}} {{\partial t}}(\tau ,x) + [A(\tau )u(\tau )](x) - V(\tau ,x)u(\tau ,x) = 0} \hfill \\ {u(t,x) = f(x),} \hfill \\ \end{array} } \right. $$ where E is a locally compact second countable Hausdorff topological space equipped with a reference measure m, f ∈ L∞(m), and V satisfies a Kato type condition. It is assumed that a transition probability density p is given, and the family of operators A(τ) is defined by $$ A(\tau )h(x) = \mathop {\lim }\limits_{^{^{\epsilon \to 0 + } } } \frac{{Y(\tau + \epsilon ,\tau )h(x) - h(x)}} {\epsilon }, $$ where Y denotes the free backward propagator associated with p. It is shown in the paper that under some restrictions on p, V , τ0 ∈ [0,t), and x0 ∈ E, the backward Feynman-Kac propagator YV associated with p and V generates a viscosity solution to the terminal value problem above at the point (τ0, x0). Similar result holds in the case where the function V is replaced by a time-dependent family μ of Borel measures on E.

Global solutions of Navier-Stokes equations with large $L^2$ norms in a new function space

First we prove certain pointwise bounds for the fundamental solutions of the perturbed linearized Navier-Stokes equation (Theorem 1.1). Next, utilizing a new framework with very little $L^p$ theory or Fourier analysis, we prove existence of global classical solutions for the full Navier-Stokes equation when the initial value has a small norm in a new function class of Kato type (Theorem 1.2). The smallness in this function class does not require smallness in $L^2$ norm. Furthermore we prove that a Leray-Hopf solution is regular if it lies in this class, which allows much more singular functions then before (Corollary 1). For instance this includes the well-known result in [25]. A further regularity condition (form boundedness) was given in Section 5. We also give a different proof about the $L^2$ decay of Leray-Hopf solutions and prove pointwise decay of solutions for the three-dimensional Navier-Stokes equations (Corollary 2, Theorem 1.2). Whether such a method exists was asked in a survey paper [2].

Single-valued extension property at the points of the approximate point spectrum

A localized version of the single-valued extension property is studied at the points which are not limit points of the approximate point spectrum, as well as of the surjectivity spectrum. In particular, we shall characterize the single-valued extension property at a point λ o∈ C in the case that λ o I− T is of Kato type. From this characterizations we shall deduce several results on cluster points of some distinguished parts of the spectrum.

Harnack’s inequality for solutions of some degenerate elliptic equations

We prove a Harnack’s inequality for non-negative solutions of some degenerate elliptic operators in divergence form with the lower order term coefficients satisfying a Kato type condition.

Continuity of eigenvalues for Schrödinger operators, $L^p$ -properties of Kato type integral operators

Given an arbitrary relatively compact (finely) open subset of ${\mathbb R}^d, \mu$ -eigenvalues of $-A(D)+\nu$ are studied where $A(D)$ is the Dirichlet Laplacian on D and $\mu,\nu$ are measures on ${\mathbb R}^d$ such that $G_X^\mu$ is continuous and $G_X^\nu$ is bounded for every ball X in ${\mathbb R}^d (G_X$ being Green's function for X). Moreover, it is shown that these eigenvalues depend continuously on D and $\nu$ . The results are based on very general compactness and convergence properties of integral operators of Kato type which are developed before.

A Kato type theorem on zero viscosity limit of Navier-Stokes flows

We present a necessary and sufficient condition for the convergence of solutions of the incompressible Navier-Stokes equations to that of the Euler equations at vanishing viscosity. Roughly speaking convergence is true in the energy space if and only if the energy dissipation rate of the viscous flows due to the tangential derivatives of the velocity in a thick enough boundary layer, a small quantity in classical boundary layer theory, approaches zero at vanishing viscosity. This improves a previous result of T. Kato (1984) in the sense that we require tangential derivatives only while the total gradient is needed in Kato’s work. However we require a slightly thicker boundary layer. We also improve our previous result where only sufficient conditions were obtained. Moreover we treat more general boundary condition which includes Taylor-Couette type flows. Several applications are presented as well. ∗wang@math.iastate.edu, 1-(515)294-1752 (T), 1-(515)294-5454(F)

On the Minimal Extension ofC0-Semigroups for Second-Order Damped Equations

The present paper deals with a minimal extension of the classical semigroup theory for second-order damped differential equations in Banach spaces with closed, densely defined linear operators as coefficients. We do not ask anymore from our operators than in the case of first-order equations, i.e., semigroups. We present here generalizations of the Miyadera–Phillips–Feller theorem, the Hille type theorem, and the Trotter–Kato type theorem. The method is quite general and could be used for equations of any order. We focus our attention on a particular dynamical operator solution or main propagator and we assume some properties about it. From this we can obtain some information about the complementary basis operator solutions or secondary propagators.

Systems of conditional differential inequalities of the Kato type

Systems of conditional differential inequalities of the Kato type

On a method of defining the Fourier transform of nonselfadjoint operators

We give a generalization of the study of nonselfadjoint perturbations of selfadjoint operators and the Fourier transforms of such perturbations to the case when the unperturbed operator is nonselfadjoint and has a simple structure. In defining and studying the Fourier transform of the perturbed operator we use smoothness of Kato type of the perturbation with respect to the unperturbed operator.

Small solutions to nonlinear Schrödinger equations

It is shown that the initial value problem for the nonlinear Schrödinger equations∂tu=iΔu+P(u,∇xu,u¯,∇xu¯),t∈ℝ,x∈ℝn,where Ρ (.) is a polynomial having no constant or linear terms, is locally well posed for a class of “small” data u0.The main ingredients in the proof are new estimates describing the smoothing effect of Kato type for the group {eitΔ}−∞∞. This method extends to systems and other dispersive models.

愛媛県で分離されたRickettsia tsutsugamushiの性状

Five strains of Rickettsia tsutsugamushi were isolated in Ehime Prefecture during December 1987 to January 1990. Of these, two strains, the Yamazaki and Noma-3, were isolated at Noma area of Imabari city and three strains, the Kakiwara-10, -11, -12, at Kakiwara area of Uwajima city. The Yamazaki strain was isolated from a patient of tsutsugamushi disease and the other strains from wild rodents (Apodemus speciosus). These strains showed virulence in euthymic mice. The calculated LD50 of the Yamazaki and Kakiwara-10 strains showed 10(-3.0) and 10(-1.8), respectively. The immunofluorescent antibody test using thirty monoclonal antibodies to six representative strains, the Gilliam, Karp, Kato, Irie, Hirano and Shimokoshi, revealed that two strains isolated at Noma area, the Yamazaki and Noma-3, were identified as the Karp type and three strains at Kakiwara area, the Kakiwara-10, -11, -12, were identified as the Kato type. It was clarified that the serotypic differences were present among the strains isolated in Ehime Prefecture. Moreover, these five strains isolated in Ehime Prefecture did not react with the serotype-specific monoclonal antibodies to the Irie, Hirano and Shimokoshi strains known as the representative strains of so-called new type of tsutsugamushi disease, showing antigenic differences.

A Kato type inequality for Riemannian submersions with totally geodesic fibers

A Kato type inequality is proved for a riemannian submersion with totally geodesic fibers. It compares such a situation with the trivial fibration and generalizes analogous results for vector bundles.

A non-self-adjoint perturbation of the continuous spectrum with smoothness of Kato type

A non-self-adjoint perturbation of the continuous spectrum with smoothness of Kato type

Haemophilus infections in chickens. 2. Types of Haemophilus paragallinarum Isolates from chickens with infectious coryza, in relation to Haemophilus gallinarum strain no. 221.

Eighty-eight Haemophilus isolates derived from chickens with infectious coryza were classified. Then they were compared biologically and serologically with Haemophilus gallinarum type I and III of strain No.221 described by Kato et al., and with H. paragallinarum strain H-18 reported previously by the authors. Of the isolates, 86 were capsulated and biologically identified as H. paragallinarum. Of them, 62 strains were hemagglutinable and serologically identical with H. gallinarum type I of strain No.221 (H. paragallinarum serotype 1) and 24 strains unhemagglutinable and serologically identical with H. paragallinarum strain H-18 (H. paragallinarum serotype 2). All the capsulated strains corresponded to either serotype 1 or 2, which showed a similar degree of pathogenicity for chickens, irrespective of serotype. The remaining 2 isolates were noncapsulated and nonpathogenic, and identical with Kato's type III strain, which was a variant derived from capsulated H. paragallinarum serotype 1.

On Perturbation of Non-Linear Equations in Banach Spaces

Condition (1) states that given /eX0 and A>0 there is a ueD(A) satisfying the equation u — lAuBf. It is also known that under condition (1) (with X0 — D(A)) A generates a contraction semigroup on D(A) (cf. [5, Theorem I]). Our first purpose is to discuss sufficient conditions for (1). In general, the direct verification of (1) is not easy. We shall give some conditions on A which implies condition (1). Our conditions seem to be weaker than (1) and hence would be easy to check. We note, however, that our conditions are, in fact, equivalent to (1). Next, given dissipative operators A and J3, we consider the perturbation problem of Kato type; in which one wants to show