Szeg\H o limit theorems for Toeplitz operators on compact homogeneous spaces
I. I.
Hirschman;
D. S.
Liang;
E. N.
Wilson
351-376
Abstract: Let $f$ be a real valued integrable function on a compact homogeneous space $M = K\backslash G$ and ${M_f}$ the operator of pointwise multiplication by $ f$. The authors consider families of Toeplitz operators ${T_{f,P}} = P{M_f}P$ as $P$ ranges over a net of orthogonal projections from $ {L^2}(M)$ to finite dimensional $G$-invariant subspaces. Necessary and sufficient conditions are given on the net in order that the distribution of eigenvalues of these Toeplitz operators is asymptotic to the distribution of values of $f$ in the sense of Szegö's classical theorem for the circle. Explicit sequences satisfying these conditions are constructed for all compact Lie groups and for all Riemannian symmetric compact spaces.
Maximal positive boundary value problems as limits of singular perturbation problems
Claude
Bardos;
Jeffrey
Rauch
377-408
Abstract: We study three types of singular perturbations of a symmetric positive system of partial differential equations on a domain $ \Omega \subset {{\mathbf{R}}^n}$. In all cases the limiting behavior is given by the solution of a maximal positive boundary value problem in the sense of Friedrichs. The perturbation is either a second order elliptic term or a term large on the complement of $\Omega$. The first corresponds to a sort of viscosity and the second to physical systems with vastly different properties in $\Omega$ and outside $\Omega$. The results show that in the limit of zero viscosity or infinitely large difference the behavior is described by a maximal positive boundary value problem in $\Omega$. The boundary condition is determined in a simple way from the system and the singular terms.
Mathematical theory of single channel systems. Analyticity of scattering matrix
I. M.
Sigal
409-437
Abstract: We show that the $ S$-matrix of a quantum many-body, short-range, single-channel system has a meromorphic continuation whose poles occur at most at the dilation-analytic resonances [28], [24] and at the eigenvalues of the Hamiltonian. In passing, we prove the main spectral theorem (on location of the essential spectrum) and asymptotic completeness for the mentioned class of systems.
Relations between $H\sp{p}\sb{u}$ and $L\sp{p}\sb{u}$ with polynomial weights
Jan-Olov
Strömberg;
Richard L.
Wheeden
439-467
Abstract: Relations between $ L_u^p$ and $H_u^p$ of the real line are studied in the case when $p > 1$ and $ u(x) = \vert q(x){\vert^p}w(x)$, where $q(x)$ is a polynomial and $w(x)$ satisfies the ${A_p}$ condition. It turns out that $ H_u^p$ and $L_u^p$ can be identified when all the zeros of $q$ are real, and that otherwise $H_u^p$ can be identified with a certain proper subspace of $L_u^p$. In either case, a complete description of the distributions in $H_u^p$ is given. The questions of boundary values and of the existence of dense subsets of smooth functions are also considered.
The group of automorphisms of a class of finite $p$-groups
Arye
Juhász
469-481
Abstract: Let $G$ be a finite $p$-group and denote by ${K_i}(G)$ the members of the lower central series of $G$. We call $G$ of type $(m,\,n)$ if (a) $G$ has nilpotency class $m - 1$, (b) $ G/{K_2}(G) \cong {{\mathbf{Z}}_{{p^n}}} \times {{\mathbf{Z}}_{{p^n}}}$ and ${K_i}(G)/{K_{i + 1}}(G) \cong {{\mathbf{Z}}_{{p^n}}}$ for every $i$, $2 \leqslant i \leqslant n - 1$. In this work we describe the structure of $\operatorname{Aut} (G)$ and certain relations between $ \operatorname{Out} (G)$ and $G$.
Differentiability of the metric projection in Hilbert space
Simon
Fitzpatrick;
R. R.
Phelps
483-501
Abstract: A study is made of differentiability of the metric projection $ P$ onto a closed convex subset $K$ of a Hilbert space $H$. When $K$ has nonempty interior, the Gateaux or Fréchet smoothness of its boundary can be related with some precision to Gateaux or Fréchet differentiability properties of $P$. For instance, combining results in $ \S3$ with earlier work of R. D. Holmes shows that $K$ has a ${C^2}$ boundary if and only if $P$ is ${C^1}$ in $ H\backslash K$ and its derivative $P'$ has a certain invertibility property at each point. An example in $\S5$ shows that if the ${C^2}$ condition is relaxed even slightly then $ P$ can be nondifferentiable (Fréchet) in $ H\backslash K$.
Nonorientable surfaces in some non-Haken $3$-manifolds
J. H.
Rubinstein
503-524
Abstract: If a closed, irreducible, orientable $3$-manifold $M$ does not possess any $2$-sided incompressible surfaces, then it can be very useful to investigate embedded one-sided surfaces in $M$ of minimal genus. In this paper such $ 3$-manifolds $ M$ are studied which admit embeddings of the nonorientable surface $ K$ of genus $3$. We prove that a $3$-manifold $M$ of the above type has at most $3$ different isotopy classes of embeddings of $K$ representing a fixed element of ${H_2}(M,\,{Z_2})$. If $M$ is either a binary octahedral space, an appropriate lens space or Seifert manifold, or if $ M$ has a particular type of fibered knot, then it is shown that the embedding of $ K$ in $M$ realizing a specific homology class is unique up to isotopy.
Fundamental groups of topological $R$-modules
Ann
Bateson
525-536
Abstract: The main result of this paper is that if $R$ is a countable, Noetherian ring, then the underlying abelian group of every $R$-module is isomorphic to the fundamental group of some topological $R$-module. As a corollary, it is shown that for certain varieties $V$(e.g., varieties of finite type) every abelian group in $V$ is isomorphic to the fundamental group of some arcwise connected topological algebra in $V$.
Measurable parametrizations of sets in product spaces
V. V.
Srivatsa
537-556
Abstract: Various parametrization theorems are proved. In particular the following is shown: Let $B$ be a Borel subset of $I \times I$ (where $I = [0,\,1]$) with uncountable vertical sections. Let $\sum \dot \cup N$ be the discrete (topological) union of $\sum$, the space of irrationals, and $ N$, the set of natural numbers with discrete topology. Then there is a map $ f:I \times (\sum \dot \cup N) \to I$ measurable with respect to the product of the analytic $\sigma$-field on $I$ (that is, the smallest $\sigma $-field on $I$ containing the analytic sets) and the Borel $\sigma$-field on $ \sum \dot \cup N$ such that $f(t,\,\, \cdot ):\,\sum \dot \cup N \to I$ is a one-one continuous map of $ \sum \dot \cup N$ onto $\{ x:(t,\,x) \in B\}$ for each $ t \in T$. This answers a question of Cenzer and Mauldin.
Saturation properties of ideals in generic extensions. I
James E.
Baumgartner;
Alan D.
Taylor
557-574
Abstract: We consider saturation properties of ideals in models obtained by forcing with countable chain condition partial orderings. As sample results, we mention the following. If $ M[G]$ is obtained from a model $M$ of GCH via any $\sigma$-finite chain condition notion of forcing (e.g. add Cohen reals or random reals) then in $ M[G]$ every countably complete ideal on $ {\omega _1}$ is ${\omega _3}$-saturated. If "$\sigma $-finite chain condition" is weakened to "countable chain condition," then the conclusion no longer holds, but in this case one can conclude that every $ {\omega _2}$-generated countably complete ideal on $ {\omega _1}$ (e.g. the nonstationary ideal) is $ {\omega _3}$-saturated. Some applications to ${\mathcal{P}_{{\omega _1}}}({\omega _2})$ are included and the role played by Martin's Axiom is discussed. It is also shown that if these weak saturation requirements are combined with some cardinality constraints (e.g. ${2^{{\aleph _1}}} > {({2^{{\aleph _0}}})^ + })$), then the consistency of some rather large cardinals becomes both necessary and sufficient.
Formal spaces with finite-dimensional rational homotopy
Yves
Félix;
Stephen
Halperin
575-588
Abstract: Let $S$ be a simply connected space. There is a certain principal fibration ${K_1} \to E\mathop \to \limits^\pi {K_0}$ in which ${K_1}$ and ${K_0}$ are products of rational Eilenberg-Mac Lane spaces and a continuous map $\phi :S \to E$ such that in particular ${\phi _0} = \pi \circ \phi $ maps the primitive rational homology of $S$ isomorphically to that of ${K_0}$. A main result of this paper is the Theorem. If $\dim \pi {}_{\ast}(S) \otimes {\mathbf{Q}} < \infty$ then $\phi$ is a rational homotopy equivalence if and only if all the primitive homology in $ H{}_{\ast}(S;\,{\mathbf{Q}})$ and $ H{}_{\ast}({K_0},\,S;\,{\mathbf{Q}})$ can (up to integral multiples) be represented by spheres and disk-sphere pairs. Corollary. If $S$ is formal, $\phi$ is a rational homotopy equivalence.
Free Lie subalgebras of the cohomology of local rings
Luchezar L.
Avramov
589-608
Abstract: A criterion is established, in terms of the Massey products structure carried by the homology of partial resolutions, for the Yoneda cohomology algebra ${\operatorname{Ext} _A}(k,\,k)$ to be a free module over the universal envelope of a free graded Lie subalgebra. It is shown that several conjectures on the (co)homology of local rings, in particular on the asymptotic behaviour of the Betti numbers, follow from such a structure. For all rings with $\operatorname{edim} A - \operatorname{depth} A \leqslant 3$, and for Gorenstein rings with $\operatorname{edim} A - \operatorname{depth} A = 4$, the following dichotomy is proved: Either $ A$ is a complete intersection, or ${\operatorname{Ext} _A}(k,\,k)$ contains a nonabelian free graded Lie subalgebra.
Hopf manifolds and spectral geometry
Kazumi
Tsukada
609-621
Abstract: We characterize Hopf manifolds in the class of Hermitian manifolds by the spectra of the real Laplacians and the complex Laplacians.
Spectral properties of orthogonal polynomials on unbounded sets
T. S.
Chihara
623-639
Abstract: We consider orthogonal polynomials when the three term recurrence formula for the monic polynomials has unbounded coefficients. We obtain information relative to three questions: Under what conditions on the coefficients will the derived set of the spectrum have a finite infimum $ \sigma$? If $ \sigma$ is finite, when will there be at most finitely many spectral points smaller than $\sigma$; and when will the distribution function be continuous at $\sigma$?
The structure of quasinormal operators and the double commutant property
John B.
Conway;
Pei Yuan
Wu
641-657
Abstract: In this paper a characterization of those quasinormal operators having the double commutant property is obtained. That is, a necessary and sufficient condition is given that a quasinormal operator $T$ satisfy the equation $\{ T\} '' = \mathcal{A}(T)$, the weakly closed algebra generated by $T$ and $1$. In particular, it is shown that every pure quasinormal operator has the double commutant property. In addition two new representation theorems for certain quasinormal operators are established. The first of these represents a pure quasinormal operator $ T$ as multiplication by $ z$ on a subspace of an $ {L^2}$ space whenever there is a vector $f$ such that $\{ \vert T{\vert^k}{T^j}f:\,k,\,j \geqslant 0\}$ has dense linear span. The second representation theorem applies to those pure quasinormal operators $T$ such that $ {T^{\ast}}T$ is invertible. The second of these representation theorems will be used to determine which quasinormal operators have the double commutant property.
Spectral permanence for joint spectra
Raul E.
Curto
659-665
Abstract: For a ${C^{\ast}}$-subalgebra $A$ of a $ {C^{\ast}}$-algebra $ B$ and a commuting $ n$-tuple $a = ({a_1}, \ldots ,{a_n})$ of elements of $ A$, we prove that $\operatorname{Sp} (a,\,A) = \operatorname{Sp} (a,\,B)$, where $ \operatorname{Sp}$ denotes Taylor spectrum. As a consequence we prove that $0 \notin \operatorname{Sp} (a,\,A)$ if and only if $\displaystyle \hat a = \left( {\begin{array}{*{20}{c}} {{d_1}} & {} & {} {d_... ...d{array} } \right) \in L\left( {A \otimes {{\mathbf{C}}^{{2^{n - 1}}}}} \right)$ is invertible, where ${d_i}$ is the $i$th boundary map in the Koszul complex for $ A$. More generally, we show that $ {\sigma _{\delta ,k}}(a,\,A) = {\sigma _{\delta ,k}}\left( {a,\,B} \right)$ and ${\sigma _{\pi ,k}}(a,\,A) = {\sigma _{\pi ,k}}(a,\,B)$ (all $k$), where ${\sigma _{\delta ,\cdot}}$ and ${\sigma _{\pi ,\cdot}}$ are the joint spectra considered by Z. Słodkowski.
The reciprocal of an entire function of infinite order and the distribution of the zeros of its second derivative
John
Rossi
667-683
Abstract: Let $f$ be a real entire function of infinite order whose zeros together with those of $f'$ are all real. It is proved that $ (1/f)''$ has an infinity of nonreal zeros. The location of the zeros of $ f''$ and $(1/f)''$ is also investigated. The result complements a finite order result of Hellerstein and Williamson.
The Bergman kernel function and proper holomorphic mappings
Steven R.
Bell
685-691
Abstract: It is proved that a proper holomorphic mapping $f$ between bounded complete Reinhardt domains extends holomorphically past the boundary and that if, in addition, $ {f^{ - 1}}(0) = \{ 0\}$, then $f$ is a polynomial mapping. The proof is accomplished via a transformation rule for the Bergman kernel function under proper holomorphic mappings.
The nonfinite generation of ${\rm Aut}(G)$, $G$ free metabelian of rank $3$
S.
Bachmuth;
H. Y.
Mochizuki
693-700
Abstract: The group of automorphisms of the free metabelian group of rank $ 3$ is not finitely generated.