Extending an operator from a Hilbert space to a larger Hilbert space,so as to reduce its spectrum

In our earlier paper [1] we showed that given any elementx of a commutative unital Banach algebraA, there is an extensionA′ ofA such that the spectrum ofx inA′ is precisely the essential spectrum ofx inA. In [2], we showed further that ifT is a continuous linear operator on a Banach spaceX, then there is an extensionY ofX such thatT extends continuously to an operatorT ⁻ onY, and the spectrum ofT ⁻ is precisely the approximate point spectrum ofT. In this paper we take the second of these results, and show further that ifX is a Hilbert space then we can ensure thatY is also a Hilbert space; so any operatorT on a Hilbert spaceX is the restriction to one copy ofX of an operatorT ⁻ onX ⊕X, whose spectrum is precisely the approximate point spectrum ofT. This result is “best possible” in the sense that if isany extension to a larger Banach space of an operatorT, it is a standard exercise that the approximate point spectrum ofT is contained in the spectrum of .     

A general theory of structure spaces with applications to spaces of prime ideals

A structure space is a quadrupleX=(X, d, A, P), where for some setR, X A=2 ^R ,d:X×X A is defined byd(I, J)=J–I, andP is the family of cofinite subsets ofR. Forr P, I X, N _r (I)={J X: d(I, J) r},To(X)={Q X: if x Q there is anr P such thatN _r (x) Q}. ThenTo(X) is a (not usually Hausdorff) topology onX called the hull-kernel topology. Replacing d byd ^*, whered ^* (I, J)=d(J, I), or byd ^s, whered ^s (I, J.)=d(I, J) d ^* (I, J), and proceeding in the obvious way yields thedual hull-kernel topology To(X ^*) andsymmetric topology To(X ^s ). The latter is always a zero-dimensional Hausdorff space. When R is a commutative ring with identity andX is a collection of proper prime ideals ofR, To(X ^s ) is usually called thepatch topology. Our generality enables us to improve on known results in the case of space of prime ideals and to apply this theory to a wide variety of algebraic structures. In particular, we establish criteria for a subspace of a structure space to be closed in the symmetric topology; we establish a duality between families of maximal elements in the hull-kernel topology and families of minimal elements in the dual hull-kernel topology of subspaces that are closed in the symmetric topology; we use topological constructions to generalize certain ring theoretic notions, such as radical ideals an annihilator ideals; we use this theory to obtain new results about subspaces of the space prime ideals of a reduced, commutative ring.Presented by F. E. J. Linton.This author's research was supported by a grant from the CUNY-PSC research award program.     

Densely two-valued metric projections in uniformly convex Banach spaces

For every uniformly convex Banach spaceX with dimX2 there is a residual setU in the Hausdorff metric spaceB(X) of bounded and closed sets inX such that the metric projection generated by a set fromU is two-valued and upper semicontinuous on a dense and everywhere continual subset ofX. For any two closed and separated subsetsM ₁ andM ₂ ofX the points on the equidistant hypersurface which have best approximations both inM ₁ andM ₂ form a dense G set in the induced topology.The author is partially supported by the National Fund for Scientific Research at the Bulgarian Ministry of Science and Education under contract MM 408/94.     

Subsets of a Hilbert space,having finite hausdorff dimension

Let X₂, X₂ be Hilbert spaces, X₂ X₁, X₂ is dense in X₁, the imbedding is compact,m X₂, dim_H ⁱ m and h⁽ⁱ⁾(m) are the Hausdorff dimension and the limit capacity (information dimension) of the setm with respect to the metrics of the spaces X_i (i=1, 2). Two examples are constructed. 1) An example of a setm bounded in X₂, such that: a) h⁽¹⁾(m) < (and, consequently, dim_H ¹ m); b)m cannot be covered by a countable collection of sets, compact in X₂ (and, consequently, dim_H ² m=). 2) an Example of a setm, compact in X₂, such that h⁽¹⁾(m) < and h⁽²⁾(m)=.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 163, pp. 154–165, 1987.     

Frobenius-Perron operators and approximation of invariant measures for set-valued dynamical systems

A set-valued dynamical systemF on a Borel spaceX induces a set-valued operatorF onM(X) — the set of probability measures onX. We define arepresentation ofF, each of which induces an explicitly defined selection ofF; and use this to extend the notions of invariant measure and Frobenius-Perron operators to set-valued maps. We also extend a method ofS. Ulam to Markov finite approximations of invariant measures to the set-valued case and show how this leads to the approximation ofT-invariant measures for transformations , whereT corresponds to the closure of the graph of .     

Classification of semispaces according to their types in infinite-dimensional vector spaces

It is shown that each semispaceC X naturally generates a relation of complete preorder onX with respect to which the pair (X C, C) is a cut ofX. By identifying the type of the semispace with the type of the cut generated by this semispace, the semispaces are classified according to their types. The obtained classification extends the classification of semispaces in finite-dimensional vector spaces due to Martinez-Legaz and Singer to infinite-dimensional spaces.Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 191–198, August, 1998.     

Normality in products with a countable factor

The class of normal spaces that have normal product with every countable space is considered. A countably compact normal space X and a countable Y such that X×Y is not normal is constructed assuming CH. Also, ? is used to construct a perfectly normal countably compact X and a countable Y such that X×Y is not normal. The question whether a Dowker space can have normal product with itself is considered. It is shown that if X is Dowker and contains any countable non-discrete subspace, then X² is not normal. It follows that a product of a Dowker space and a countable space is normal if and only if the countable space is discrete. If X is Rudin's ZFC Dowker space, then X² is normal. An example of a Dowker space of cardinality ℵ₂ with normal square is constructed assuming .     

A note on sub-multiplicative norms

Summary IfX is a finite-dimensional linear space andL(X) the linear space of linear operators onX thenL(X) may be represented asXX ^*. IfE={e ₁, ...,e _n } is a basis forX and e _j y _j ^* is a typical element ofXX ^*, then norms can be introduced onL(X) in the form y _j ^* e _j . Given that the norm onX isE-absolute we derive a necessary and sufficient condition for the norm onL(X) to be submultiplicative.     

Wijsman topology on function spaces

LetC(X,Y) be the space of continuous functions from a metric space (X,d) to a metric space (Y, e).C(X, Y) can be thought as subset of the hyperspaceCL(X×Y) of closed and nonempty subsets ofX×Y by identifying each element ofC(X,Y) with its graph. We considerC(X,Y) with the topology inherited from the Wijsman topology induced onCL(X×Y) by the box metric ofd ande. We study the relationships between the Wijsman topology and the compact-open topology onC(X,Y) and also conditions under which the Wijsman topology coincide with the Fell topology. Sufficient conditions under which the compactopen topology onC(X,Y) is weaker than the Wijsman topology are given (IfY is totally bounded, then for every metric spaceX the compactopen topology onC(X,Y) is weaker than the Wijsman topology and the same is true forX locally connected andY rim-totally bounded). We prove that a metric spaceX is boundedly compact iff the Wijsman topology onC(X, ℝ) is weaker than the compact-open topology. We show that ifX is a σ-compact complete metric space andY a compact metric space, then the Wijsman topology onC(X,Y) is Polish.     

A Reuniformization for Contractive Mappings in Uniform Spaces

Let X be a complete uniform HAUSDORFF space with a uniformity generated by a saturated family of pseudometrics ?? = {?_α(x, y): α ? A} and let T: X → X be a continuous mapping. The paper contains necessary and sufficient conditions for the existence of a new family of pseudometrics ??*={?*(x, y): α*?A*} generated the same topology such that T is contractive with respect to ??*.     

Operator stable laws: Series representations and domains of normal attraction

LetX,X ₁,X ₂,... be i.i.d. random vectors in ^d. The limit laws that can arise by suitable affine normalizations of the partial sums,S _n=X ₁+...+X _n, are calledoperator-stable laws. These laws are a natural extension to ^d of the stable laws on. Thegeneralized domain of attraction of [GDOA()] is comprised of all random vectorsX whose partial sums can be affinely normalized to converge to . If the linear part of the affine transformation is restricted to take the formn ^–B for some exponent operatorB naturally associated to thenX is in thegeneralized domain of normal attraction of [GDONA()]. This paper extends the theory of operator-stable laws and their domains of attraction and normal attraction.     

High density limit theorems for nonlinear chemical reactions with diffusion

Summary The solutionX of a nonlinear reaction-diffusion equation on then-dimensional unit cubeS is approximated by a space-time jump Markov processX _v,N (law of large numbers (LLN)).X _v,N is constructed on a gridS _N onS ofN cells, wherev is proportional to the initial number of particles in each cell. The deviation ofX _v,N fromX is computed by a central limit theorem (CLT). The assumptions on the parametersv, N are for the LLN: , asN , and for the CLT: , asN . The limitY =Y ^X in the CLT, which is a generalized Ornstein-Uhlenbeck process, is represented as the mild solution of a linear stochastic partial differential equation (SPDE) and its best possible state spaces are described. The problem of stationary solutions ofY ^X in dependence ofX is also investigated.On leave from Universität Bremen. This work was supported by the Stiftung Volkswagenwerk and a grant from ONR     

Homomorphisms onC ∞ (E) andC ∞-bounding sets

We prove, for the class of real locally convex spacesE that are continuously and linearly injectable into somec ₀(), that every non-zero homomorphism on the algebraC (E) ofC -functions onE is given by a point evaluation at some point ofE. Furthermore, if every real-valuedC -function on the weak topology of a quasi-complete locally convex spaceE is bounded on a subsetA ofE, thenA is relatively weakly compact.     

An eigenvalue problem for the numerical range of a bounded linear operator

LetX be a complex Lebesgue space with a unique duality mapJ fromX toX ^*, the conjugate space ofX. LetA be a bounded linear operator onX. In this paper we obtain a non-linear eigenvalue problem for (A)=sup{Re: W(A} whereW(A)={J(x)A(x)) : x=1}, under the assumption that (A) and the convex hull ofW(A) for some linear operatorsA onl _p , 2<p<.     

Compact spaces, elementary submodels, and the countable chain condition, II

Given a space 〈X,T〉 in an elementary submodel of H(θ), define XM to be X∩M with the topology generated by . It is established that if XM is compact and satisfies the countable chain condition, while X is not scattered and has cardinality less than the first inaccessible cardinal, then X=XM. If the character of XM is a member of M, then “inaccessible” may be replaced by “1-extendible”.     

Polynomial Szemerédi theorems for countable modules over integral domains and finite fields

Given a pair of vector spacesV andW over a countable fieldF and a probability spaceX, one defines apolynomial measure preserving action ofV onX to be a compositionT o ϕ, where ϕ:V→W is a polynomial mapping andT is a measure preserving action ofW onX. We show that the known structure theory of measure preserving group actions extends to polynomial actions and establish a Furstenberg-style multiple recurrence theorem for such actions. Among the combinatorial corollaries of this result are a polynomial Szemerédi theorem for sets of positive density in finite rank modules over integral domains, as well as the following fact:Let be a finite family of polynomials with integer coefficients and zero constant term. For any α>0, there exists N ∈ ℕ such that whenever F is a field with |F|≥N and E ⊆F with |E|/|F|≥α, there exist u∈F, u≠0, and w∈E such that w+ϕ(u)∈E for all ϕ∈ . The first two authros are supported by NSF, grant DMS-0070566 and DMS-0245350. The second author was supported by the A. Sloan Foundation. The third author is supported by NSF, grant DMS-0200700.     

Boundedness and continuity of additive and convex functionals

Summary It is shown that for any real Baire topological vector spaceX the set classesA(X):={T : for any open and convex setD T, every Jensen-convex functional, defined onD and bounded from above onT, is continuous} andB(X):={T : every additive functional onX, bounded from above onT, is continuous} are equal. This generalizes a result of Marcin E. Kuczma (1970) who has shown the equalityA( ⁿ )=B( ⁿ ) However, the infinite dimensional case requires completely different methods; therefore, even in the caseX = ⁿ we obtain a new (and perhaps simpler) proof than that given by M. E. Kuczma.     

Bott-Chern currents,excess normal bundles and the Chern character

LetY andY be two complex submanifolds of a complex manifoldX, and assume thatU=YY is also a submanifold ofX. Let Ñ be the excess normal bundle toU inX. We attach certain Bott-Chern currents to holomorphic Hermitian vector bundles onY, Y and to resolutions of the corresponding sheaves by holomorphic Hermitian complexes overX. We show that these currens verify natural functorial properties. Our results extend earlier results of Bismut-Gillet-Soulé to the case where Ñ is nonzero.     

Discrepancy and approximations for bounded VC-dimension

Let (X, ) be a set system on ann-point setX. For a two-coloring onX, itsdiscrepancy is defined as the maximum number by which the occurrences of the two colors differ in any set in . We show that if for anym-point subset the number of distinct subsets induced by onY is bounded byO(m ^d) for a fixed integerd, then there is a coloring with discrepancy bounded byO(n ^1/2–1/2d(logn)^1+1/2d ). Also if any subcollection ofm sets of partitions the points into at mostO(m ^d) classes, then there is a coloring with discrepancy at mostO(n ^1/2–1/2dlogn). These bounds imply improved upper bounds on the size of -approximations for (X, ). All the bounds are tight up to polylogarithmic factors in the worst case. Our results allow to generalize several results of Beck bounding the discrepancy in certain geometric settings to the case when the discrepancy is taken relative to an arbitrary measure.Work of J.M. and E.W. was partially supported by the ESPRIT II Basic Research Actions Program of the EC under contract no. 3075 (project ALCOM). L.W. acknowledges support from the Deutsche Forschungsgemeinschaft under grant We 1265/1-3, Schwerpunktprogramm Datenstrukturen und effiziente Algorithmen.     

On the Fell topology

Let 2 ^X denote the closed subsets of a Hausdorff topological space <X, {gt}>. The Fell topology _F on 2 ^X has as a subbase all sets of the form {A 2 ^X :A V 0}, whereV is an open subset ofX, plus all sets of the form {A 2 ^X :A W}, whereW has compact complement. The purpose of this article is two-fold. First, we characterize first and second countability for _F in terms of topological properties for . Second, we show that convergence of nets of closed sets with respect to the Fell topology parallels Attouch-Wets convergence for nets of closed subsets in a metric space. This approach to set convergence is highly tractable and is well-suited for applications. In particular, we characterize Fell convergence of nets of lower semicontinuous functions as identified with their epigraphs in terms of the convergence of sublevel sets.