Description of Hyperinvariant Subspaces of a Contraction in Terms of Its Characteristic Function

If T is a completely nonunitary contraction on a Hilbert space and L is its invariant subspace corresponding to a regular factorization of its characteristic function = , then L is hyperinvariant if and only if the following two conditions are fulfilled: (1) supp _* supp is of Lebesgue measure zero; (2) for every pair A H (E E) and A _* H (E _* E _*) intertwining by , i.e., such that A =A _*, there exists a function A _F H (F F) intertwining with A by and with A _* by , i.e., such that A = A _F and A _F = A _*. Bibliography: 4 titles.     

On some algebras diagonalized by M-bases of ℓ2

We study reflexive algebrasA whose invariant lattices LatA are generated by M-bases of ². Examples are given whereA differs from ( being the rank one subalgebra ofA), and where together with the identity I is not strongly dense inA. For M-bases in a special class, we characterize the cases when they are strong, and also when the identity I is the ultraweak limit of a sequence of contractions in . We show that this holds provided that I is approximable by compact operators inA at any two points of ². We show that the spaceA+^* (where is the annihilator of ) is ultraweakly dense in (²), and characterize the M-bases in this class for which the sum is direct. We give a class of automorphisms ofA which are strongly continuous but not spatial.     

Some Landau's type inequalities for infinitesimal generators

Summary Lett T(t) be a strongly continuous contraction semigroup on a complex Banach space and letA be its infinitesimal generator. We prove that, forx D(A ³), the following inequalities hold true: Ax³ 243/8 x²A ³ x, A ² x 24 xA ³ x². Ift T(t) is a contraction group (resp. cosine function) we get the analogous but better inequalities with constants 9/8 and 3 (resp. 81/40 and 72/25) instead of 243/8 and 24. We consider also uniformly bounded semigroups, groups and cosine functions.     

The intuitionistic propositional calculus with quantifiers

Let L be the language of the intuitionistic propositional calculus J completed by the quantifiers and , and let calculus 2J in language L contain, besides the axioms of J, the axioms xB (x) B(y) and B(y) xB (x). A Kripke semantics is constructed for 2J and a completeness theorem is proven. A result of D. Gabbay is generalized concerning the undecidability of C2J⁺-extension of 2J by schemes x (x B) and x(A B(x))A xB (x) specificially: the undecidability is proven of each T theory in language L such that [2J]T [C2J+] ([2J] ([2J] denotes the set of all theorems of calculus 2J).Translated from Matematicheskie Zametki, Vol. 22, No. 1, pp. 69–76, July, 1977.     

Compressibility effect on laminar convection to a radiating gas past a vertical plate

Summary Fluid motion generated by a long vertical hot plate in the presence of an externally applied magnetic field is investigated when radiative transfer is important. Even though the velocities involved are small, this paper takes compressibility into effect owing to the extreme heat encountered in radiation problems. Thus the viscosity () and the thermal conductivity (K) are assumed to vary with the temperature (T). In this paper we takeT ⁿ while the Prandtl number (Pr) which is defined in terms of the specific heat at constant pressure (c _p) byPr=c _p/K is constant. Assuming a general differential approximation for the radiative flux, the asymptotic problem, valid far away from the tips of the plate, is reduced to a set of nonlinear coupled ordinary differntial equations. Solution to this problem is discussed quantitatively.

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Zusammenfassung Es wird die Bewegung eines Fluids entlang einer heißen vertikalen Platte untersucht in Gegenwart eines außen angebrachten Magnetfeldes, im Falle daß die Strahlung wichtig ist. Obwohl die Geschwindigkeiten klein sind, werden in dieser Arbeit Kompressibilitätseffekte berücksichtigt, da bei Strahlungsproblemen hohe Temperaturen auftreten. Die Zähigkeit und der WärmeleitungskoeffizientK werden mit der Temperatur veränderlich angenommen, mitT ⁿ, und der Prandtl-ZahlPr=c _p/K konstant. Für den Strahlungsfluß wird eine allgemeine différentielle Form angenommen, und das asymptotische Problem, gültig in einigem Abstand von den Plattenenden, wird auf ein System von nichtlinearen gekoppelten gewöhnlichen Differentialgleichungen reduziert. Die Lösung des Problems wird quantitativ diskutiert.

     

On the cohomology of a hyperfinite action

LetV be an ergodic automorphism of a probability space and letA be a locally compact abelian group. Iff:XA is a measurable function, one defines a skew product automorphismV _f of the product spaceX×A byV _f (x,)=(Vx,+f(x)) for every point (x,) inX×A. In this paper we prove thatV _f is ergodic for most functionsf.     

The Ramsey property for collections of sequences not containing all arithmetic progressions

A family of sequences has the Ramsey property if for every positive integerk, there exists a least positive integerf (k) such that for every 2-coloring of {1,2, ...,f (k)} there is a monochromatick-term member of . For fixed integersm > 1 and 0 q < m, let _q(m) be the collection of those increasing sequences of positive integers {x ₁,..., x_k} such thatx _i+1 – x_i q(modm) for 1 i k – 1. Fort a fixed positive integer, denote byA _t the collection of those arithmetic progressions having constant differencet. Landman and Long showed that for allm 2 and 1 q < m, _q(m) does not have the Ramsey property, while _q(m) A _m does. We extend these results to various finite unions of _q(m) 's andA _t 's. We show that for allm 2, _q=1 ^m–1 _q(m) does not have the Ramsey property. We give necessary and sufficient conditions for collections of the form _q(m) ( _{t T} A _t) to have the Ramsey property. We determine when collections of the form _a(m₁) _b(m₂) have the Ramsey property. We extend this to the study of arbitrary finite unions of _q(m)'s. In all cases considered for which has the Ramsey property, upper bounds are given forf .     

A note on rational approximation rate for Müntz system {x λn } with λ n ↘ 0

[Zho2] {x ⁿ } , _n 0 n .

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Supported in part by an NSERC Postdoctoral Fellowship and a CRF grant of University of Alberta.     

On extremal perturbations of semi-Fredholm operators

Given a bounded linear operatorA in an infinite dimensional Banach space and a compact subset of a connected component of its semi-Fredholm domain, we construct a finite rank operatorF such that –A+F is bounded below (or surjective) for each ,F ²=0 and rankF=max min{dimN(–A), codimR(–A)}, if ind(–A)0 (or ind(–A)0, respectively) for each .     

On the spectrum determined growth assumption and the perturbation ofC 0 semigroups

The spectrum determined growth property ofC ₀ semigroups in a Banach space is studied. It is shown that ifA generates aC ₀ semigroup in a Banach spaceX, which satisfies the following conditions: 1) for any >s(A), sup{R(;A) | Re}<; 2) there is a ₀>(A) such that , xX, and , fX ^*, then (A=s(A). Moreover, it is also shown that ifA=A ₀+B is the infinitesimal generator of aC ₀ semigroup in Hilbert space, whereA ₀ is a discrete operator andB is bounded, then (A)=s(A). Finally the results obtained are applied to wave equation and thermoelastic system.     

Embeddings of Lorentz—Marcinkiewicz spaces with mixed norms

X(Y) f -:X(Y)={fM(×): f_{X(Y)=f(x,.)YX<} . =(0, ), M (×) — , ×, X, Y, Z— . X(Y) Z(×).     

Bounded projections and duality on spaces of holomorphic functions in the unit ball ofℂ n

In this paper three Banach spacesA ₀(),A andA ¹() of functions holomorphic in the unit ballB of ⁿ are defined. We exhibit bounded projections fromC ₀(B) ontoA ₀(), fromL ¹(B) ontoA ¹(), and fromL(B) ontoA(). Using these projections, we show thatA ₀()* A ¹() andA ¹()* A().Supported in part by the National Natural Science Foundation of China.     

Geometric properties of 2-dimensional Brownian paths

Summary Let A be the set of all points of the plane , visited by 2-dimensional Brownian motion before time 1. With probability 1, all points of A are twist points except a set of harmonic measure zero. Twist points may be continuously approached in \A only along a special spiral. Although negligible in the sense of harmonic measure, various classes of cone points are dense in A, with probability 1. Cone points may be approached in \A within suitable wedges.Research supported in part by NSF Grant DMS 8419377     

A remark on normal derivations of Hilbert-Schmidt type

LetB (H) denote the algebra of operators on the separable Hilbert spaceH. LetC ₂ denote the (Hilbert) space of Hilbert-Schmidt operators onH, with norm .₂ defined by S ₂ ² =(S,S)=tr(SS ^*). GivenA, B B (H), define the derivationC (A, B):B(H)B(H) byC(A, B)X=AX-XB. We show that C(A,B)X+S ₂ ² =C(A,B)X ₂ ² +S ₂ ² holds for allXB(H) and for everySC ₂ such thatC(A, B)S=0 if and only if reducesA, ker S reducesB, andA | S and B| ker S are unitarily equivalent normal operators. We also show that ifA, BB(H) are contractions andR(A, B)B(H)B(H) is defined byR(A, B)X=AXB-X, thenSC ₂ andR(A, B)S=0 imply R(A,B)X+S ₂ ² =R(A,B)X ₂ ² +S ₂ ² for allXB(H).     

Functional calculus and its application to operator algebras

In this paper we will prove that any unital isometric representation ofH () with finitely connected domain has property (A ₁(r)) for somer1, which generalizes the same conclusion in [2] and [8].     

Theorem of Bartle,Dunford, and Schwartz concerning vector measures

We show the existence, for an arbitrary vector measure: x (where X is a Banach space and gs is a-algebra of subsets of a set S) of a functional x X (X is the conjugate space of X) such that is absolutely continuous with respect to _x, _x (E)=(E)>, E gs.Translated from Matematicheskie Zametki, Vol. 7, No. 2, pp. 247–254, February, 1970.     

Regularity of ∫Ω|∇u|2 + λ∫Ω|u−f{2 and some gap phenomenon

We study the regularity of the minimizer u for the functional F (u,f)=|u|² + |u–f{² over all maps uH ¹(, S ²). We prove that for some suitable functions f every minimizer u is smooth in if ₀ and for the same functions f, u has singularities when is large enough.

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Résumé On étudie la régularité des minimiseurs u du problème de minimisation min_ueH ¹(,S²)(|u|² + |u–f{². On montre que pour certaines fonctions f, u est régulière lorsque ₀ et pour les mêmes f, si est assez grand, alors u possède des singularités.

     

Weight functions on the Kneser graph and the solution of an intersection problem of Sali

LetX, Y be finite sets and suppose thatF is a collection of pairs of sets (F, G),FX,GY satisfying |FF|s, |GG|t and |FF|+|GG|s+t+1 for all (F, G),F, GF. Extending a result of Sali, we determine the maximum ofF.     

Adjoints of slant Toeplitz operators

In this paper, we will use the Birkhoff's ergodic theorem to do some finer analysis on the spectral properties of slant Toeplitz operators. For example, we will show that if is an invertibleL function on the unit circle, then almost every point in (A ^* ) is not an eigenvalue ofA ^* . More specifically, we will show that the point spectrum ofA ^* is contained in a circle with positive radius.     

Quasi-coherence of Hardy spaces in several complex variables

In this paper, we prove that the Hardy spaceH ^p (), 1p<, over a strictly pseudoconvex domain in ⁿ with smooth boundary is quasi-coherent. More precisely, we show that Toeplitz tuplesT with suitable symbols onH ^p () have property (). This proof is based on a well known exactness result for the tangential Cauchy-Riemann complex.