A Spectral Condition for Strong Compactness

The evolutionary Schrödinger equation with a second-order generator is considered on the line. For the Schrödinger equation with a degenerate operator, the characteristic form of which vanishes outside a segment I = [−l, l] ⊂ ℝ, a well-posed setting of the Cauchy problem is investigated. We find conditions for the initial-value data of the problem that are necessary and sufficient for its unique solvability in a given interval of time. A sequence of regularized Cauchy problems with uniformly elliptic operators is considered as well; we study the convergence of the sequence of solutions for nondegenerate problems to the solution of the degenerate problem as well as the convergence of regularized semigroups of transformations in the strong operator topology. We prove that any arbitrary sequence of solutions of regularized problems with initial data that does not satisfy the existence condition for the solution diverges. However, one cannot exclude that there exists a subsequence of the regularization parameters such that the corresponding sequence of regularized semigroups converges (in the strong operator topology) uniformly on each segment. We provide a description of the set of all possible partial limits for the sequence of regularized semigroups; this description is given in terms of a collection of self-adjoint extensions for the degenerate operator. It is still an open question if all those partial limits are accessible. The Cauchy problem for the Schrödinger equation, the generator of which is a symmetric linear differential operator in the Hilbert space H = L 2(ℝ), is considered as well. We investigate if the behavior of the sequence of regularized semigroups depends on the choice of the regularization for the generator. We define a linear self-adjoint regularization of the Cauchy problem with a degenerate operator as a directed set of well-posed problems approximating the original one. We define a correct regularization as a linear self-adjoint regularization of the degenerate operator such that its index of error determines the well-posedness and the convergence and weak convergence of the sequence of regularized solutions. We find necessary and sufficient conditions for the convergence (in the strong and weak operator topologies) of the sequence of correctly regularized semigroups.

Let B(X) be the operator algebra for a separable infinite dimensional Hilbert space H, endowed with the strong operator topology or *-strong topology. We give sufficient conditions for a continuous linear mapping L: B(X) → B(X) to be supercyclic or *-supercyclic. In particular our condition implies the existence of an infinite dimensional subspace of supercyclic vectors for a mapping T on H. Hypercyclicity of the operator algebra with strong operator topology was studied by Chan and here we obtain an analogous result in the case of *-strong operator topology.

We consider a singularity perturbed nonlinear differential equation εu ′ =f(x)u++εP(x,u,ε) which we suppose real analytic for x near some interval [a,b] and small |u|, |ε|. We furthermore suppose that 0 is a turning point, namely that xf(x) is positive if x≠0. We prove that the existence of nicely behaved (as ϵ→0) local (at x=0) or global, real analytic or C ∞ solutions is equivalent to the existence of a formal series solution ∑u n (x)ε n with u n analytic at x=0. The main tool of a proof is a new "principle of analytic continuation" for such "overstable" solutions. We apply this result to the second order linear differential equation εy ′′ +φ(x,ε)y ′ +ψ(x,ε)y=0 with φ and ψ real analytic for x near some interval [a,b] and small |ε|. We assume that -xφ(x,0) is positive if x≠0 and that the function ψ 0 :x↦ψ(x,0) has a zero at x=0 of at least the same order as φ 0 ↦φ(x,0). For this equation, we prove that the existence of local or global, real analytic or C ∞ solutions tending to a nontrivial solution of the reduced equation φ(x,0)y ′ +ψ(x,0)y=0 is equivalent to the existence of a non trivial formal series solution y ^(x,ε)=∑y n (x)ε n with y n analytic at x=0. This improves and generalizes a result of C.H. Lin on this so-called " Ackerberg-O’Malley resonance" phenomenon. In the proof, the problem is reduced to the preceding problem for the corresponding Riccati equation In the final section, we construct examples of such second order equations exhibiting resonance such that the formal solution y ^ has a prescribed logarithmic derivative y ^ ′ (0,ε)/y ^(0,ε) at x=0 which is divergent of Gevrey order 1.

Many of the fundamental research problems in the geometry of normed linear spaces can be loosely phrased as: Given a Banach space X and a class of Banach spaces Y does X contain a subspace Y ∈ Y? As a Banach space X is determined by its unit ball B x ≡ { x ∈ X :‖ x ‖ ≤ 1 } the problem can be rephrased in terms of the geometry of convex sets: Can a given unit ball B x be sliced with a subspace to obtain a set in some given class of unit balls? A result of this type is the famous theorem of Dvoretzky (see also [L], [M6], [M4], [MS], [FLM]).

The purpose of this note is to examine the following issues related to the Chernoff estimate: (1) For contractions on a Banach space, we modify the [Formula: see text]-estimate and apply it in the proof of the Chernoff product formula for [Formula: see text]-semigroups in the strong operator topology. (2) We use the idea of a probabilistic approach, proving the Chernoff estimate in the strong operator topology to uplift it to the operator-norm estimate for quasi-sectorial contraction semigroups. (3) The operator-norm Chernoff estimate is applied to quasi-sectorial contraction semigroups for proving the operator-norm convergence of the Dunford–Segal approximants.

LetX be a Banach Space and letB(X) denote the family of bounded linear operators onX. LetR+ = [0, ∞). A one parameter family of operators {S(t);t ∈R+},S:R+ →B(X), is called exponential-cosine operator function ifS(O) =I andS(s +t) − 2S(s)S(t) = (S(2s) − 2S2(s))S(t −s), for alls, t ∈R+,s ≦t. Let\(Af = \mathop {\lim }\limits_{h \to 0} \frac{{S(h)f - f}}{h}\),f∈D(A), and\(Bf = \mathop {\lim }\limits_{h \to 0} \frac{{S(2h)f - 2S(h)f + f}}{{h^2 }}\),f∈D(B). It is shown that for a strongly continuous exponential-cosine operator {S(t)},f∈D(A2) implies ι0t(t −u(S(u)fdu∈D(B) and B ι0t(t −u)S(u)fdu =S(t)f −f +tAf − 2A ι0tS(u)fdu + 2A2 ι0t(t −u)S(u)fdu.D(B) is seen to be dense inD(A2). Some regularity properties ofS(t) have also been obtained.

Classical results on weakly compactly generated (WCG) Banach spaces imply the existence of projectional resolutions of identity (PRI) and the existence of many projections on separable subspaces (SCP). We address the questions if these can be the only projections in a nonseparable WCG space, in the sense that there is a PRI (Pα : ω ≤ α ≤ λ) such that any projection is the sum of an operator in the closure of the linear span of countably many Pα's (in the strong operator topology) and a separable range operator. Wark's modification of Shelah's and Steprāns' construction provides an unconditional example for λ = ω1. We note that it is impossible for λ > ω2.

A rigorous solution is presented for a problem associated with a circular inclusion embedded within an infinite matrix in plane elastostatics. The bonding at the inclusion-matrix interface is assumed to be imperfect. Specifically, the jump in the normal displacement is assumed to be proportional to the normal traction with the proportionality parameter taken to be circumferentially inhomogeneous. In addition, we assume that displacements in the tangential direction are continuous. This type of interface is generally referred to as an inhomogeneous non-slip interface. Using the principle of analytic continuation, the basic boundary-value problem for four analytic functions is reduced to a first-order differential equation for a single analytic function defined inside the circular inclusion. The resulting closed-form solutions include a finite number of unknown constants determined by analyticity requirements and certain other supplementary conditions. The method is illustrated using several specific examples of a particular class of inhomogeneous non-slip interface. The results from these calculations are compared with the corresponding results when the interface imperfections are homogeneous. These comparisons indicate that the circumferential variation of interface damage has a significant effect on even the average stresses induced within a circular inclusion.

The infrared freezing of observables is known to hold at fixed orders of perturbative QCD if the Minkowskian quantities are defined through the analytic continuation from the Euclidean region. In a recent paper [1] it is claimed that infrared freezing can be proved also for Borel resummed all-orders quantities in perturbative QCD. In the present paper we obtain the Minkowskian quantities by the analytic continuation of the all-orders Euclidean amplitudes expressed in terms of the inverse Mellin transform of the corresponding Borel functions [2]. Our result shows that if the principle of analytic continuation is preserved in Borel-type resummations, the Minkowskian quantities exhibit a divergent increase in the infrared regime, which contradicts the claim made in [1]. We discuss the arguments given in [1] and show that the special redefinition of Borel summation at low energies adopted there does not reproduce the lowest order result obtained by analytic continuation.

In a host of applications in engineering, it is necessary to obtain information about a system over a broad range. In most cases it is not possible to evaluate the parameter of interest in closed form. However, either theoretical or experimental data is available in a narrow band. Generation of the data over the broadband is not possible or may be extremely time consuming. The principle of analytic continuation is utilized to extrapolate/interpolate the data over a wide band. The method of Cauchy [1] has been chosen in this paper to carry out the analytic continuation.

In this paper we present a characterization of the symmetric rotational periodic gravity waterwaves of finite depth and without stagnation points in terms of the underlying flow.Namely, we show that such a wave is symmetric and has a single crest and trough per period if and only ifthere exists a vertical line within the fluid domain such that allthe fluid particles located on that line minimize there simultaneously their distance to the fluid bedas they move about. Our analysis uses the moving plane method, sharp elliptic maximum principles, and the principle of analytic continuation.

For the duality mapping on a Banach space the relation between lower semi-continuity and upper semi-continuity properties is explored, upper semi-continuity is characterized in terms of slices of the ball and upper semi-continuity properties are related to geometrical properties which imply that the space is an Asplund space. The duality mapping is a natural set-valued mapping from the unit sphere of a normed linear space into subsets of its dual sphere, and which for an inner product space is the mapping associating an element of the unit sphere with the corresponding continuous linear functional given by the inner product. It is an example of a subdifferential mapping of a continuous convex function (in this case, the norm), which is in turn a special kind of maximal monotone mapping. Cudia [4, p. 298] showed that the duality mapping is always upper semi-continuous when the space has the norm and the dual space has the weak* topology, and Kenderov [10, p. 67] extended this to maximal monotone mappings. Bonsall, Cain, and Schneider [3] used the property to prove the connectedness of the numerical range of an operator on a normed linear space. Along with the activity which culminated in StegalΓs theorem [15] characterizing an Asplund space as one whose dual has the Radon-Nikodym property, there has been some interest in finding geometrical conditions sufficient for a space to be Asplund. A Banach space X is an Asplund space if every continuous convex function defined on an open convex subset of X is strongly differentiate on a dense Gδ subset of its domain. Ekeland and Lebourg [6, p. 204] have shown that a Banach space is Asplund if there exists a strongly differentiate real function on the space with bounded nonempty support, in particular, if the space can be equivalently renormed to have norm strongly differentiable on the unit sphere. Using StegalΓs theorem, a result of Diestel and Faires [5, p. 625] gives that a Banach space is Asplund if the space can be equivalently renormed to be very smooth, that is, to be smooth and to have the single valued duality mapping continuous when the space has the norm and the dual space has the weak topology. Recently Smith and Sullivan [13, Theorem 15] have exhibited a more general condition which is sufficient for a Banach space to be Asplund. We show that

It is shown that in every complex Banach space E with open unit ball D C E there is a closed C-linear subspace Cc such that V C\ D is the orbit of the origin 0 E E under the group Aut(D) of all biholomorphic automorphisms of D. In particular two complex Banach spaces are isometrically equivalent if and only if their open unit balls are biholomorphically equivalent.Suppose E, F are complex Banach spaces and U E E is an open subset.Then a mapping/: U -* Fis called holomorphic if the Frchet derivative of/ at x (denoted by f'(x)) exists for every x G U. (For details compare [6].)In this paper we show that the Banach space E is completely determined by the holomorphic structure of its (open) unit ball; more precisely, Theorem.The complex Banach spaces E and are isometrically isomorphic if and only if there is a biholomorphic mapping from the unit ball of E onto the unit ball of .This answers a question L. A. Harris raised in [3, p. 15].For the proof let be a fixed complex Banach space with unit ball DEE.Let 0 denote the set of all holomorphic vector fields on D i.e. of all expressions X = f(z)d/dz where/: D -* E is a holomorphic mapping and z denotes the coordinate of D. Then D is a complex Lie algebra in the product [/W,^)]=(.'W/(z)-/'(z)g(z)).For every integer v the Lie algebra D contains the linear subspace ^v = l h(z)~~ h: E ^> E is a. homogeneous polynomial of degree v + 1 >.Obviously [$r,$J C SJL^ for every v and \i.Since D is a bounded domain in E, the set g of all complete holomorphic vector fields on D has the following properties (compare [7]):(0) Every h(z)d/dz G g is uniquely determined by h(0) and h'(0) (Cartan's uniqueness theorem).

【For a bounded operator T = $U{\mid}T{\mid}$ (polar decomposition), we consider a transform b $\widehat{T}$ = ${\mid ${\mid}T{\mid}U$ and discuss the convergence of iterated transform of $\widehat{T}$ under the strong operator topology. We prove that such iteration of quasiaffine hyponormal operator converges to a normal operator under the strong operator topology.】

Let $E$ and $F$ be locally convex spaces and $G$ their completed $\varepsilon$-tensor product. It is shown that if $S$ and $T$ are weakly almost periodic equicontinuous semigroups of operators on $E$ and $F$ respectively, then, under mild restrictions on $E$ or $F, S \otimes T$ is a weakly almost periodic equicontinuous semigroup of operators on $G$, and the almost periodic and flight vector subspaces of $G$ are related in a natural way to the corresponding subspaces of $E$ and $F$ via the $\varepsilon$-tensor product. Furthermore, if $E$ and $F$ both decompose into a direct sum of these subspaces then so does $G$.