Two characterization theorems for integral operators

Let (X,) be a separable -finite measure space. A bounded operator A on L²(X) is called an integral operator if it is induced by an equation: Af(x) = k(x,y)f(y)d(y), where k is a measurable function on X × X such that |k(x,y)f(y)|d(y) < a.e. for every f in L²(X).In this paper, some results on Carleman operators, due to von Neumann, Tarjonski and Weidmann, are extended to the case of the general integral operator.     

On measures that agree on balls

The fundamental result: if and v are two finite Borel measures, defined in the spaceL ^p[0, 1] (1p<) or in C(K) (K is a metric compactum without isolated points), then from the equalities (B)=v(B) for all balls B of radius 1 there follows that =v. In addition, in the spaces C(K) and _p (1p<) from the inequalities (B) v(B) there follows that v.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 177, pp. 122–128, 1989.     

Non uniqueness of invariant means for amenable group actions

It is shown, that for the action of a -compact group, being amenable as an abstract discrete group, on a locally compact measure space (X, , ), is not the unique invariant mean. Furthermore, this paper gives a characterisation of probability spaces, having a unique invariant mean for the action of an amenable group.     

Jensen inequalities for functions with higher monotonicities

Summary We investigate generalizations of the classical Jensen and Chebyshev inequalities. On one hand, we restrict the class of functions and on the other we enlarge the class of measures which are allowed. As an example, consider the inequality (J)(f(x) d) A (f(x) d, d d = 1. Iff is an arbitrary nonnegativeL _x function, this holds if 0, is convex andA = 1. Iff is monotone the measure need not be positive for (J) to hold for all convex withA = 1. If has higher monotonicity, e.g., is also convex, then we get a version of (J) withA < 1 and measures that need not be positive.     

Uniform boundedness of a family of set functions

Let be a ring of sets, X a normed space, : X ( ) a bounded family of triangular functions. The following generalized Nikodym theorem is established: the family {} is uniformly bounded on if and only if it is bounded on every sequence of pairwise disjoint sets of which the union is a part of some set in . An analogous criterion is established also for semiadditive functions. In addition, it is shown that uniform boundedness of a family of triangular functions is preserved in passing from a ring to the -ring it generates.Translated from Matematicheskie Zametki, Vol. 23, No. 6, pp. 855–861, June, 1978.     

Finitely subadditive and countably subadditive outer measures

Results are given comparing countably subadditive (csa) outer measures and finitely subadditive (fsa) outer measures, especially relating to regularity and measurability conditions such as (*) condition:A setE (of an arbitrary setX), is measurable ( an outer measure),ES (the collection of measurable sets) iff (X)=(E)+(E). Specific examples are given contrasting csa and fsa outer measures. In particular fsa and csa outer measures derived from finitely additive measures defined on an algebra of sets generated by a lattice of sets, are investigated in some detail.     

The existence of probability measures with specified projections

Let X and Y be locally compact-compact topological spaces, F X×Y is closed, and P(F) is the set of all Borel probability measures on F. For us to find, for the pair of probability measures (_x, _y P (X)×P(Y), a probability measure P(F) such that _X = _X ^–1 , _Y = _Y ^–1 it is necessary and sufficient that, for any pair of Borel sets A X, B Y for which (A× B) F=Ø, the condition _XA+ _YB 1 holds.Translated from Matematicheskie Zametki, Vol. 14, No. 4, pp. 573–576, October, 1973.     

Excesses of systems of exponential functions

Conditions on the closeness of real sequences {_n} and {_n} are studied which imply the equality of the excesses of the systems {exp(i_nx)} and {exp(i_nx)} in the space L²(–a, a). A theorem is formulated in terms of the difference of the sequences {_n} and {_n} enumerating the functions. In the corollaries of the theorem, conditions are given in terms of the behavior of the difference _n–_n0. An example is constructed showing that the condition _n–_n0 alone is not sufficient for equality of the excesses.Translated from Matematicheskie Zametki, Vol. 22, No. 6, pp. 803–814, December, 1977.     

Quasianapole Interaction of Charged Fermions

We obtain the analytic expression for the total cross section of the reaction e ^– e ⁺l ^– l ⁺ (l=,) taking possible quasianapole interaction effects into account. We find numerical restrictions on the interaction parameter value from data for the reaction e ^– e ^+–+ in the energy domain below the Z ⁰ peak.     

Approximate Identities in Spaces of all Absolutely Continuous Measures on Locally Compact Semigroups

Let G be a locally compact group. Then M_a (G), the space of all absolutely continuous measures on G, has a bounded approximate identity. Baker and Baker proved that (S) (the space of all measures M(S) so that maps x _x *|| and x ||*_x are weak continuous from a locally compact semigroup S into M(S)) is closed under absolutely continuity and has an approximate identity. The main purpose of this paper is to show that similar results hold true for a locally compact semigroup S and M_a(S) the space of all absolutely continuous measures on S.     

Durch graduierte Lie-Algebren definierte Paar-Algebren

Pair algebras which have a non degenerate (left- and right-) invariant bilinear form and for which the inner derivation algebra is completely reducible are characterised by pairs (C,), where C is a n×n matrix satisfying certain conditions and is a sequence of n integers equal to 0 or 1. They occur as pair algebras of type (S(C,)_–1,S(C,)₁), xuy=[[x,u],y], where (S(C,)_r)_r is the gradation induced by . in the Kac-Moody algebraS(C). If C is an affin Cartan matrix (as in the case of Lie triple systems), there exists a finite dimensional simple Lie algebrag and a Aut (g), ord =m< such that the pair algebra is isomorphic to the pair algebra (g _–1,g ₁), xuy=[[x,u],y] (product ing), whereg _i. is the eigenspace of of eigenvalue ⁱ, a primitive m-th root of unity.     

The martin boundary for harmonic functions on groups of automorphisms of a homogeneous tree

Consider a closed subgroup of the automorphism group of a homogeneous treeT, and assume that acts transitively on the vertex set. Suppose that is a probability measure on which has continuous density with respect to Haar measure and whose support is compact open and generates as a closed semigroup. It is shown that the Martin boundary of with respect to the random walk with law coincides with the space of ends ofT. This extends known results for free groups and applies, for example, to the affine group over a non archimedean local field.     

Some extensions of the Hewitt-Savage zero-one law

Summary Let denote the class of infinite product probability measures = ₁× ₂× defined on an infinite product of replications of a given measurable space (X, A), and let denote the subset of for which (A) =0 or 1 for each permutation invariant event A. Previous works by Hewitt and Savage, Horn and Schach, Blum and Pathak, and Sendler (referenced in the paper) discuss very restrictive sufficient conditions under which a given member , of belongs to . In the present paper, the class is shown to possess several closure properties. E.g., if and _{0 n} for some n 1, then ₀× ₁× ₂×.... While the current results do not permit a complete characterization of they demonstrate conclusively that is a much larger subset of than previous results indicated. The interesting special case X={0,1} is discussed in detail.Research supported by the National Science Foundation under grant No. MCS75-07556     

Zum Einbettungsproblem für Markoff-Prozesse

Every elementary Markov process with a polish state space and with a discrete set of time parameter dense in ₊, whose finite dimensional distributions are derived from a semigroup (K _t) of Markov kernels continuous in 0 ₊ and whose initial distribution satisfies K _t, can be imbedded in an elementary Markov process with the same state space and with parameter set ₊ so that the corresponding finite dimensional distributions are equal.     

Some special results on convergent sequences of radon measures

Two problems will be considered. In Part I we consider a class of subsets of a topological space X and a Radon measure on X; if it is known that, for sufficiently many , the restrictions of the sets in constitutes a uniformity class in T w.r.t. the restriction of the given measure, then we ask if it follows that is a uniformity class in X.Part II, which can be read independently of Part I, is concerned with the question whether, to a given convergent sequence of Radon measures, say _n, there always exist sufficiently many compact sets K such that _n(K)(K).     

Stochastic integrals on general topological measurable spaces

Summary A general theory of stochastic integral in the abstract topological measurable space is established. The martingale measure is defined as a random set function having some martingale property. All square integrable martingale measures constitute a Hilbert space M ². For each M ², a real valued measure on the predictable -algebra is constructed. The stochastic integral of a random function with respect to is defined and investigated by means of Riesz's theorem and the theory of projections. The stochastic integral operator I is an isometry from L ²() to a stable subspace of M ², its inverse is defined as a random Radon-Nikodym derivative. Some basic formulas in stochastic calculus are obtained. The results are extended to the cases of local martingale and semimartingale measures as well.     

A New Relaxation Space for Obstacles

The space of obstacles (i.e. p-quasi upper semicontinuous functions) is endowed with a distance which is topologically equivalent to the -convergence. We find the metric completion of this space and we give some application for minimization problems of cost functionals depending on obstacles via their level sets. An element of the completion is a decreasing and _p -continuous on the left mapping Rt _t , where _t are positive Borel measures vanishing on sets of zero p-capacity.     

Orders of one-sided approximations of functions inL p -metric

. L _p , 0<p<, . , f, {E _n (f) _p } ₁ p>0 .

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The author expresses his thanks to S. B. Stekin for the attention he has paid to this work.     

Outlines for the computation of the multiplicities of the spectra of orthogonal sums

If A and B are operators in the spaces X and Y, respectively, and if the operator B has many sets , , such that the manifolds p is a polynomial are dense in the space Y, then Here a=(the multiplicity of the spectrum of the operator A)=mindimL: span (AⁿL:n0)}=X. For example, if B=T^–g is a Toeplitz operator in the space H² with antianalytic symbol) and if g (the polynomial convex hull of the spectrum (A)) , then. Conversely, if and, then (under some assumptions on the regularity of the function f we have. One also gives examples of univalent and essentially univalent functions f (f H), for which Tf>1.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 126, pp. 150–158, 1983.     

On the almost everywhere convergence of ergodic averages for power-bounded operators on LP-subspaces

Let X be a closed subspace of L^P(), where is an arbitrary measure and 1A(n) and (n) denote the discrete ergodic averages and Hilbert transform truncates defined by U. We extend to this setting the -a. e. convergence criteria forA(n) and (n) which V. F. Gaposhkin and R. Jajte introduced for unitary operators on L²(). Our methods lift the setting from X to ^p, where classical harmonic analysis and interpolation can be applied to suitable square functions.