The diagonal entries in the formula ``quasitriangular $-$ compact $=$ triangular'' and restrictions of quasitriangularity
Domingo A.
Herrero
1-42
Abstract: A (bounded linear) Hilbert space operator $T$ is called quasitriangular if there exists an increasing sequence $\{ {P_n}\} _{n = 0}^\infty $ of finite-rank orthogonal projections, converging strongly to 1, such that $\left\Vert {(1 - {P_n})T{P_n}} \right\Vert \to 0\,(n \to \infty )$. This definition, due to P. R. Halmos, plays a very important role in operator theory. The core of this article is a concrete answer to the following problem: Suppose $T$ is a quasitriangular operator and $\Gamma = \{ {\lambda _j}\} _{j = 1}^\infty$ is a sequence of complex numbers. Find necessary and sufficient conditions for the existence of a compact operator $K$ (of arbitrarily small norm) so that $T - K$ is triangular with respect to some orthonormal basis, and the sequence of diagonal entries of $T - K$ coincides with $\Gamma$. For instance, if no restrictions are put on the norm of $K$, then $T$ and $\Gamma$ must be related as follows: (a) if ${\lambda _0}$ is a limit point of $\Gamma$ and ${\lambda _0} - T$ is semi-Fredholm, then $ {\operatorname{ind}}({\lambda _0} - T) > 0$; and (b) if $\Omega$ is an open set intersecting the Weyl spectrum of $T$, whose boundary does not intersect this set, then $ \{ j:{\lambda _j} \in \Omega \}$ is a denumerable set of indices. Particularly important is the case when $\Gamma = \{ 0,0,0, \ldots \}$. The following are equivalent for an operator $T$: (1) there is an integral sequence $\{ {P_n}\} _{n = 0}^\infty$ of orthogonal projections, with rank ${P_n} = n$ for all $n$, converging strongly to 1, such that $\left\Vert {(1 - {P_n})T{P_{n + 1}}} \right\Vert \to 0\,(n \to \infty )$; (2) from some compact $ K,\,T - K$ is triangular, with diagonal entries equal to 0; (3) $T$ is quasitriangular, and the Weyl spectrum of $T$ is connected and contains the origin. The family ${({\text{StrQT}})_{ - 1}}$ of all operators satisfying (1) (and hence (2) and (3)) is a (norm) closed subset of the algebra of all operators; moreover, $ {({\text{StrQT}})_{ - 1}}$ is invariant under similarity and compact perturbations and behaves in many senses as an analog of Halmos's class of quasitriangular operators, or an analog of the class of extended quasitriangular operators $ {({\text{StrQT}})_{ - 1}}$, introduced by the author in a previous article. If $ \{ {P_n}\} _{n = 0}^\infty$ is as in (1), but condition $\left\Vert {(1 - {P_n})T{P_{n + 1}}} \right\Vert \to 0\,(n \to \infty )$ is replaced by (1') $ \left\Vert {(1 - {P_{{n_k}}})T{P_{{n_k} + 1}}} \right\Vert \to 0\,(k \to \infty )$ for some subsequence $\{ {n_k}\} _{k = 1}^\infty $, then (1') is equivalent to (3'), $T$ is quasitriangular, and its Weyl spectrum contains the origin. The family ${({\text{QT}})_{ - 1}}$ of all operators satisfying (1') (and hence (3')) is also a closed subset, invariant under similarity and compact perturbations, and provides a different analog to Halmos's class of quasitriangular operators. Both classes have ``$ m$-versions'' ( $ {({\text{StrQT}})_{ - m}}$ and, respectively, ${({\text{QT}})_{ - m}}$, $m = 1,2,3, \ldots$) with similar properties. ( $ {({\text{StrQT}})_{ - m}}$ is the class naturally associated with triangular operators $A$ such that the main diagonal and the first $ (m - 1)$ superdiagonals are identically zero, etc.) The article also includes some applications of the main result to certain nest algebras ``generated by orthonormal bases.''
Decay with a rate for noncompactly supported solutions of conservation laws
Blake
Temple
43-82
Abstract: We show that solutions of the Cauchy problem for systems of two conservation laws decay in the supnorm at a rate that depends only on the ${L^1}$ norm of the initial data. This implies that the dissipation due to increasing entropy dominates the nonlinearities in the problem at a rate depending only on the ${L^1}$ norm of the initial data. Our results apply to any BV initial data satisfying ${u_0}( \pm \infty ) = 0$ and $ {\operatorname{Sup}}\{ {u_0}( \cdot )\} \ll 1$. The problem of decay with a rate independent of the support of the initial data is central to the issue of continuous dependence in systems of conservation laws because of the scale invariance of the equations. Indeed, our result implies that the constant state is stable with respect to perturbations in $ L_{{\operatorname{loc}}}^1$. This is the first stability result in an $ {L^p}$ norm for systems of conservation laws. It is crucial that we estimate decay in the supnorm since the total variation does not decay at a rate independent of the support of the initial data. The main estimate requires an analysis of approximate characteristics for its proof. A general framework is developed for the study of approximate characteristics, and the main estimate is obtained for an arbitrary number of equations.
Surgery on codimension one immersions in ${\bf R}\sp {n+1}$: removing $n$-tuple points
J. Scott
Carter
83-101
Abstract: The self-intersection sets of immersed $n$-manifolds in $(n + 1)$-space provide invariants of the $ n$th stable stem and the $ (n + 1)$st stable homotopy of infinite real projective space. Theorems of Eccles [5] and others [1, 8, 14, 19] relate these invariants to classically defined homotopy theoretic invariants. In this paper a surgery theory of immersions is developed; the given surgeries affect the self-intersection sets in specific ways. Using such operations a given immersion may be surgered to remove $ (n + 1)$-tuple and $ n$-tuple points, provided the $ {\mathbf{Z}}/2$-valued $ (n + 1)$-tuple point invariant vanishes $(n \geq 5)$. This invariant agrees with the Kervaire invariant for $ n = 4k + 1$. These results first appeared in my dissertation [2]; a summary was presented in [3]. Some results and methods have been improved since these works were written. In particular, the proof of Theorem 14 has been simplified.
On generalizing Boy's surface: constructing a generator of the third stable stem
J. Scott
Carter
103-122
Abstract: An analysis of Boy's immersion of the projective plane in $ 3$-space is given via a collection of planar figures. An analogous construction yields an immersion of the $3$-sphere in $4$-space which represents a generator of the third stable stem. This immersion has one quadruple point and a closed curve of triple points whose normal matrix is a $ 3$-cycle. Thus the corresponding multiple point invariants do not vanish. The construction is given by way of a family of three dimensional cross sections.
Frobenius reciprocity and extensions of nilpotent Lie groups
Jeffrey
Fox
123-144
Abstract: In $\S1$ we use $ {C^\infty }$-vector methods, essentially Frobenius reciprocity, to derive the Howe-Richardson multiplicity formula for compact nilmanifolds. In $\S2$ we use Frobenius reciprocity to generalize and considerably simplify a reduction procedure developed by Howe for solvable groups to general extensions of nilpotent Lie groups. In $\S3$ we give an application of the previous results to obtain a reduction formula for solvable Lie groups.
Affine semigroups and Cohen-Macaulay rings generated by monomials
Ngô Viêt
Trung;
Lê Tuân
Hoa
145-167
Abstract: We give a criterion for an arbitrary ring generated by monomials to be Cohen-Macaulay in terms of certain numerical and topological properties of the additive semigroup generated by the exponents of the monomials. As a consequence, the Cohen-Macaulayness of such a ring is dependent upon the characteristic of the ground field.
Polar classes and Segre classes on singular projective varieties
Shoji
Yokura
169-191
Abstract: We investigate the relation between polar classes of complex varieties and the Segre class of $K$. Johnson [Jo]. Results are obtained for hypersurfaces of projective spaces and for certain varieties with isolated singularities.
Index filtrations and the homology index braid for partially ordered Morse decompositions
Robert
Franzosa
193-213
Abstract: On a Morse decomposition of an invariant set in a flow there are partial orderings defined by the flow. These are called admissible orderings of the Morse decomposition. The index filtrations for a total ordering of a Morse decomposition are generalized in this paper with the definition and proof of existence of index filtrations for admissible partial orderings of a Morse decomposition. It is shown that associated to an index filtration there is a collection of chain complexes and chain maps called the chain complex braid of the index filtration. The homology index braid of the corresponding admissible ordering of the Morse decomposition is obtained by passing to homology in the chain complex braid.
Simplexwise linear untangling
David W.
Henderson
215-226
Abstract: In this paper we show how to canonically untangle simplexwise linear spanning arcs of a convex $2$-cell. Specifically, we show that the space of such arcs is contractible. The main step in the contraction is a flow along the gradient field of an energy function. A $3$-dimensional version of this result would imply the Smale Conjecture--Hatcher Theorem.
Morse theory for codimension-one foliations
Steven C.
Ferry;
Arthur G.
Wasserman
227-240
Abstract: It is shown that a smooth codimension-one foliation on a compact simply-connected manifold has a compact leaf if and only if every smooth real-valued function on the manifold has a cusp singularity.
Dense imbedding of test functions in certain function spaces
Michael
Renardy
241-243
Abstract: In a recent paper [1], J. U. Kim studies the Cauchy problem for the motion of a Bingham fluid in ${R^2}$. He points out that the extension of his results to three dimensions depends on proving the denseness of ${C^\infty }$-functions with compact support in certain spaces. In this note, such a result is proved.
Rational Moore $G$-spaces
Peter J.
Kahn
245-271
Abstract: This paper obtains some existence and uniqueness results for Moore spaces in the context of the equivariant homotopy theory of Bredon. This theory incorporates fixed-point-set data as part of the structure and so is a refinement of the classical equivariant homotopy theory. To avoid counterexamples to existence in the classical case and to focus on new phenomena involving the fixed-point-set structure, most of the results involve rational spaces. In this setting, there are no obstacles to existence, but a notion of projective dimension presents an obstacle to uniqueness: uniqueness is proved, subject to constraint on the projective dimension, and an example shows that this constraint is sharp. Various related existence results are proved and computations are given of certain equivariant mapping sets $ [X,\,Y]$, $X$ an equivariant Moore space.
Equivariant homology decompositions
Peter J.
Kahn
273-287
Abstract: This paper presents some results on the existence of homology decompositions in the context of the equivariant homotopy theory of Bredon. To avoid certain obstructions to the existence of equivariant Moore spaces occurring already in classical equivariant homotopy theory, most of the work of this paper is done ``over the rationals.'' The standard construction of homology decompositions by Eckmann and Hilton can be followed in the present equivariant context until it is necessary to produce appropriate $k'$-invariants. For these, the Eckmann-Hilton construction uses a certain Universal Coefficient Theorem for homotopy sets. The relevant extension of this to the equivariant situation is an equivariant Federer spectral sequence, which is developed in $\S2$. Using this, we can formulate conditions which imply the existence of the desired $ k'$-invariants, and hence the existence of the homology decomposition. The conditions involve a certain notion of projective dimension. For one application, equivariant homology decompositions always exist when the group has prime order.
On Brownian excursions in Lipschitz domains. I. Local path properties
Krzysztof
Burdzy;
Ruth J.
Williams
289-306
Abstract: A necessary and sufficient condition is given for a Brownian excursion law in a Lipschitz domain to share the local path properties with an excursion law in a halfspace. This condition is satisfied for all boundary points of every ${C^{1,\alpha }}$-domain, $ \alpha > 0$. There exists a ${C^1}$-domain such that the condition is satisfied almost nowhere on the boundary. A probabilistic interpretation and applications to minimal thinness and boundary behavior of Green functions are given.
Locally geodesically quasiconvex functions on complete Riemannian manifolds
Takao
Yamaguchi
307-330
Abstract: In this article, we investigate the topological structure of complete Riemannian manifolds admitting locally geodesically quasiconvex functions, whose family includes all geodesically convex functions. The existence of a locally geodesically quasiconvex function is equivalent to the existence of a certain filtration by locally convex sets. Our argument contains Morse theory for the lower limit function of a given locally geodesically quasiconvex function.
Simply-connected $4$-manifolds with a given boundary
Steven
Boyer
331-357
Abstract: Let $M$ be a closed, oriented, connected $ 3$-manifold. For each bilinear, symmetric pairing $({{\mathbf{Z}}^n},\,L)$, our goal is to calculate the set $ {\mathcal{V}_L}(M)$ of all oriented homeomorphism types of compact, $ 1$-connected, oriented $ 4$-manifolds with boundary $ M$ and intersection pairing isomorphic to $ ({{\mathbf{Z}}^n},\,L)$. For each pair $ ({{\mathbf{Z}}^n},\,L)$ which presents $ {H_ \ast }(M)$, we construct a double coset space $B_L^t(M)$ and a function $c_L^t:{\mathcal{V}_L}(M) \to B_L^t(M)$. The set $ B_L^t(M)$ is the quotient of the group of all link-pairing preserving isomorphisms of the torsion subgroup of ${H_1}(M)$ by two naturally occuring subgroups. When $ ({{\mathbf{Z}}^n},\,L)$ is an even pairing, we construct another double coset space ${\hat B_L}(M)$, a function ${\hat c_L}:{\mathcal{V}_L}(M) \to {\hat B_L}(M)$ and a projection ${p_2}:{\hat B_L}(M) \to B_L^t(M)$ such that $ {p_2} \cdot {\hat c_L} = c_L^t$. Our main result states that when $({{\mathbf{Z}}^n},\,L)$ is even the function ${\hat c_L}$ is injective, as is the function $ c_L^t \times \Delta :{\mathcal{V}_L}(M) \to B_L^t(M) \times {\mathbf{Z}}/2$ when $ ({{\mathbf{Z}}^n},\,L)$ is odd. Here $\Delta$ is a Kirby-Siebenmann obstruction to smoothing. It follows that the sets ${\mathcal{V}_L}(M)$ are finite and of an order bounded above by a constant depending only on $ {H_1}(M)$. We also show that when ${H_1}(M;{\mathbf{Q}}) \cong 0$ and $({{\mathbf{Z}}^n},\,L)$ is even, $c_L^t = {p_2} \cdot {\hat c_L}$ is injective. It seems likely that via the functions $c_L^t \times \Delta$ and ${\hat c_L}$, the sets $B_L^t(M) \times {\mathbf{Z}}/2$ and ${\hat B_L}(M)$ calculate ${\mathcal{V}_L}(M)$ when $({{\mathbf{Z}}^n},\,L)$ is respectively odd and even. We verify this in several cases, most notably when $ {H_1}(M)$ is free abelian. The results above are based on a theorem which gives necessary and sufficient conditions for the existence of a homeomorphism between two $1$-connected $4$-manifolds extending a given homeomorphism of their boundaries. The theory developed is then applied to show that there is an $m > 0$, depending only on ${H_1}(M)$, such that for any self-homeomorphism $f$ of $M$, ${f^m}$ extends to a self-homeomorphism of any $ 1$-connected, compact $ 4$-manifold with boundary $ M$.
The chromatic number of Kneser hypergraphs
N.
Alon;
P.
Frankl;
L.
Lovász
359-370
Abstract: Suppose the $ r$-subsets of an $ n$-element set are colored by $t$ colors. THEOREM 1.1. If $n \geq (t - 1)(k - 1) + k \cdot r$, then there are $k$ pairwise disjoint $r$-sets having the same color. This was conjectured by Erdös $[{\mathbf{E}}]$ in 1973. Let $T(n,\,r,\,s)$ denote the Turán number for $ s$-uniform hypergraphs (see $\S1$). THEOREM 1.3. If $\varepsilon > 0,\,t \leq (1 - \varepsilon )T(n,\,r,\,s)/(k - 1)$, and $ n > {n_0}(\varepsilon ,\,r,\,s,\,k)$, then there are $k$ $r$-sets ${A_1},{A_2}, \ldots ,{A_k}$ having the same color such that $\left\vert {{A_i} \cap {A_j}} \right\vert < s$ for all $ 1 \leq i < j \leq k$. If $ s = 2,\,\varepsilon$ can be omitted. Theorem 1.1 is best possible. Its proof generalizes Lovász' topological proof of the Kneser conjecture (which is the case $ k = 2$). The proof uses a generalization, due to Bárány, Shlosman, and Szücs of the Borsuk-Ulam theorem. Theorem 1.3 is best possible up to the $\varepsilon$-term (for large $n$). Its proof is purely combinatorial, and employs results on kernels of sunflowers.
On the Cauchy problem associated with the motion of a Bingham fluid in the plane
Jong Uhn
Kim
371-400
Abstract: This paper discusses an initial value problem for the variational inequality which describes the motion of a Bingham fluid in the plane. The existence of strong solution is established.
Large-time behavior of solutions to a scalar conservation law in several space dimensions
Patricia
Bauman;
Daniel
Phillips
401-419
Abstract: We consider solutions of the Cauchy problem in ${\mathbf{R}}_ + ^{n + 1}$ for the equation ${u_t} + {\operatorname{div}_x}f(u) = 0$. The initial data is assumed to be a compact perturbation of a function of the form, $\varphi (x) = a$ for $\left\langle {x,\,\mu } \right\rangle > 0$, $\varphi (x) = b$ for $\left\langle {x,\,\mu } \right\rangle < 0$, where $a$ and $b$ are constants and $\mu$ is a given unit vector. The Cauchy problem together with an entropy condition on $u$ is known to be well posed. The solution with unperturbed initial data, $\varphi (x)$, is a traveling shock, $\varphi (x - \overrightarrow k t)$, provided that $\varphi (x - \overrightarrow k t)$ satisfies the entropy condition (an inequality on $a,\,b,\,\mu$, and $f$). Assuming this type of condition on $ \varphi$, we study the large-time behavior of $u$. In particular, we show that $u$ converges to a traveling shock whose profile agrees with $\left\langle {x,\,\mu } \right\rangle = 0$ outside of a compact set.
Fixed sets of framed $G$-manifolds
Stefan
Waner
421-429
Abstract: This note describes restrictions on the framed bordism class of a framed manifold $Y$ in order that it be the fixed set of some framed $ G$-manifold $M$ with $G$ a finite group. These results follow from a recently proved generalization of the Segal conjecture, and imply, in particular, that if $M$ is a framed $G$-manifold of sufficiently high dimension, and if $G$ is a $p$-group, then the number of ``noncancelling'' fixed points is either zero or approaches infinity as the dimension of $M$ goes to infinity. Conversely, we give sufficient conditions on the framed bordism class of a manifold $ Y$ that it be the fixed set of some framed $G$-manifold $M$ of arbitrarily high dimension.