Embeddings of Banach Spaces Into Banach Lattices and the Gordon–Lewis Property

In this paper we first show that if X is a Banach space and is a left invariant crossnorm on lX, then there is a Banach lattice L and an isometric embedding J of X into L, so that I J becomes an isometry of lX onto l_m J(X). Here I denotes the identity operator on l and l_m J(X) the canonical lattice tensor product. This result is originally due to G. Pisier (unpublished), but our proof is different. We then use this to prove the main results which characterize the Gordon–Lewis property GL and related structures in terms of embeddings into Banach lattices.     

The interpolation ofℓ r sequences by hp functions

The sequence spaceH ^P (z)={{f (z_h)}:f H ^p} is defined for a fixed sequence Z={z_k} of different points of the open unit disk and the Hardy class HP of analytic functions in the disk. For an arbitrary p[1, ) is constructed a point sequence Z= {z_k} such that ¹h ^p(z), but ^r h^p (Z) for r > 1. It follows from a well-known result of L. Carleson that the inclusions ^r h (Z) for all r[1,] are equivalent.Translated from Matematicheskie Zametki, Vol. 21, No. 4, pp. 503–508, April, 1977.     

Über einen Banachraum analytischer Funktionen

Let be a Banach-space of functions f analytical in a convex domain G with derivations f(z)0 in G and sup|arg f(z)|<. The distribution of the univalent, of the bounded and of the limit-continuous functions in B will be inquired.     

The geometric structure of Chebyshev sets in ℓ<Superscript>∞</Superscript>(<Emphasis Type="Italic">n</Emphasis>)

A subset M of a normed linear space X is called a Chebyshev set if each x X has a unique nearest point in M. We characterize Chebyshev sets in (n) in geometric terms and study the approximative properties of sections of Chebyshev sets, suns, and strict suns in (n) by coordinate subspaces.__________Translated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 39, No. 1, pp. 1–10, 2005Original Russian Text Copyright © by A. R. AlimovSupported by RFBR grant No.02-01-00248.Translated by A. R. Alimov     

Remarks on ergodicity of stationary irreducible transient Markov chains

Summary Consider a stationary process {X _n(), – < n < . If the measure of the process is finite (the measure of the whole sample space finite), it is well known that ergodicity of the process {X _n(), - < n < and of each of the subprocesses {X _n(), 0 n < , {X _n(), – < n 0 are equivalent (see [3]). We shall show that this is generally not true for stationary processes with a sigma-finite measure, specifically for stationary irreducible transient Markov chains. An example of a stationary irreducible transient Markov chain {X _n(), - < n <} with {itX_n(), 0 n < < ergodic but {X _n(), < n 0 nonergodic is given. That this can be the case has already been implicitly indicated in the literature [4]. Another example of a stationary irreducible transient Markov chain with both {X _n(), 0 n < and {itX _n(),-< < n 0} ergodic but {X _n(), - < n < nonergodic is presented. In fact, it is shown that all stationary irreducible transient Markov chains {X _n(), - < n < < are nonergodic.This research was supported in part by the Office of Naval Research.John Simon Guggenheim Memorial Fellow.     

Toeplitz operators with symbols from in the spaces

It is shown that the algebra of the multipliers of the space ^p (1<<) contains the closed subalgebra C_p+H _p , which coincides with the Douglas algebra C + H for =2. It is proved that a Toeplitz operator with symbol from C_p+H _p is Fredholm on ^p if and only if its symbol is invertible in C_p+H _p .Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 157, pp. 124–128, 1987.The authors are grateful to V. I. Vasyunin for assistance.     

Characterization,structure and analysis on abelian ℒ∞ groups

An abelian topological group is an group if and only if it is a locally -compactk-space and every compact subset in it is contained in a compactly generated locally compact subgroup. Every abelian groupG is topologically isomorphic to G ₀ where ₀ andG ₀ is an abelian group where every compact subset is contained in a compact subgroup. Intrinsic definitions of measures, convolution of measures, measure algebra,L ₁-algebra, Fourier transforms of abelian groups are given and their properties are studied.     

On the equation of similar profiles

Summary Considerf+ ff+ (1–f²)+ f=0 together with the boundary conditionsf(0)=f(0)=0,f ()=1. If=–1,>0, arbitrary there is at least one solution which satisfies 0<f<1 on (0, ). By the additional conditionf>0 on (0, ) or, alternately 0<1, the uniqueness of the solution is demonstrated.If=1,<0, arbitrary the existence of solutions for which –1<f<0 in some initial interval (0,t) and satisfying generallyf>1 is established. In both problems, bounds forf (0) and qualitative behavior of the solutions are shown.

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Sommario Si consideri il problema definito dall'equazionef+ f f+ (1–f²)+ f=0 e dalle condizioni al contornof(0)=f (0)=0,f()=1. Assumendo=–1,>0, arbitrario si dimostra che esiste almeno una soluzione che soddisfa 0<f<1 nell'intervallo (0, ). Se in aggiunta si ipotizzaf>0 in (0, ), oppure 0<=1, l'unicità délia soluzione è assicurata.Successivamente si considéra il problema di valori al contorno con=1,<0, arbitrario. In questo caso esiste un'intera classe di soluzioni che soddisfano –1<f<0 in un intorno dell'origine e tali chef>1, in generale.Di detti problemi viene studiato il comportamento délle soluzioni e vengono determinate dalle maggiorazioni e minorazioni del valoref(0).

     

Über eine Verallgemeinerung der Nicoletti-Aufgabe für Funktional-Differentialgleichungen mit voreilendem Argument

We consider a functional differential equation (1) (t)=F(t,) fort[0,+) together with a generalized Nicoletti condition (2)H()=. The functionF: [0,+)×C ⁰[0,+)B is given (whereB denotes the Banach space) and the value ofF (t, ) may depend on the values of (t) fort[0,+);H: C ⁰[0,+)B is a given linear operator and B. Under suitable assumptions we show that when the solution :[0,+)B satisfies a certain growth condition, then there exists exactly one solution of the problem (1), (2).     

Subdivision algorithms with nonnegative masks generally converge

In this paper, a subdivision scheme consists of an operator froml () tol () determined by a doubly infinite sequence, called the mask. This operator convolutes, in a certain sense, sequences l () with the mask, thus producing a new sequence inl (). Moreover, this new sequence is placed on a finer grid. If we iterate this process with a positive mask infinitely many times, it is known that this process will produce a continuous function, which we callf . In this paper, we consider the extent to which non-negative masks yield similar results. An important application of subdivision schemes in computer graphics is the generation of curves and surfaces from an initial sequence.     

The Relations between Volume Ratios and New Concepts of GL Constants

In this paper we investigate a property named GL(p,q) which is closely related to the Gordon-Lewis property. Our results on GL(p,q) are then used to estimate volume ratios relative to _p, 1相似文献   

Some functional inequalities and their Baire category properties

Summary In the paper we consider, from a topological point of view, the set of all continuous functionsf:I I for which the unique continuous solution:I – [0, ) of(f(x)) (x, (x)) and(x, (x)) (f(x)) (x, (x)), respectively, is the zero function. We obtain also some corollaries on the qualitative theory of the functional equation(f(x)) = g(x, (x)). No assumption on the iterative behaviour off is imposed.     

On the Growth of the Maximum of the Modulus of an Entire Function on a Sequence

Let M _f(r) and _f(r) be, respectively, the maximum of the modulus and the maximum term of an entire function f and let be a continuously differentiable function convex on (–, +) and such that x = o((x)) as x +. We establish that, in order that the equality be true for any entire function f, it is necessary and sufficient that ln (x) = o((x)) as x +.     

Hulls for Various Kinds of α-Completeness in Archimedean Lattice-Ordered Groups

Within Archimedean -groups, and with an infinite cardinal or , we consider X-hulls where X stands for any of the following classes of -groups: -projectable; laterally -complete; boundedly laterally -complete; conditionally -complete; combinations of the preceding, together with divisibility and/or relative uniform completeness. All these hulls exist, and may be obtained by iterated adjunction of the required extra elements, within the essential hull. When the -groups is relatively -complemented one step in the iteration suffices for several crucial properties. We derive from the above a considerable number of equations involving combinations of these hull operators.     

Some complements and corrections to my papers on the theory of attractors for abstract semigroups

In Sec. 1 a correction is given of the estimate of the Hausdorff dimension and an estimate of the fractal dimension of a bounded subset of a Hilbert space, semiinvariant with respect to a flattening transformation. In Sec. 2 the results, proved by the author for semigroups with a continuous group parameter tR⁺[0, ), are carried over to the case when t runs through the semigroup ⁺{tt0} of some additive group R=(–, ).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 182, pp. 102–112, 1990.     

Finite Representability of ℓp in Quotients of Banach Spaces

A typical result of the paper states that if X is a Banach space with a basis and for some 1pq, the spaces _p and _q are finitely block representable in every block subspace of X, then every block subspace of X admits a block quotient Z such that for every r[p,q], the space _r is finitely block representable in Z. Results of a similar nature are also established for ^N _p-block-sequences and asymptotic spaces.     

Boundary-value problems for two-dimensional canonical systems

The two-dimensional canonical systemJy=–Hy where the nonnegative Hamiltonian matrix functionH(x) is trace-normed on (0, ) has been studied in a function-theoretic way by L. de Branges in [5]–[8]. We show that the Hamiltonian system induces a closed symmetric relation which can be reduced to a, not necessarily densely defined, symmetric operator by means of Kac' indivisible intervals; of. [33], [34]. The formal defect numbers related to the system are the defect numbers of this reduced minimal symmetric operator. By using de Branges' one-to-one correspondence between the class of Nevanlinna functions and such canonical systems we extend our canonical system from (0, ) to a trace-normed system on which is in the limit-point case at ±. This allows us to study all possible selfadjoint realizations of the original system by means of a boundaryvalue problem for the extended canonical system involving an interface condition at 0.     

A quantitative refinement of Rado's theorem

The fundamental result of the paper is the following. Theorem: Let be a k-quasiconformal Jordan curve and let be another Jordan curve (not necessarily quasiconformal). Assume that f maps conformallyext ontoext , f()=, f()>0. We assume that there exists a homeomorphism between and such that Then there exist numbers =(k)>0 and A=A(k), such that f(())– A, .Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 157, pp. 103–112, 1987.     

Behavior on the real line of entire functions represented by Dirichlet series

Conditions are found under which for an entire function f represented by a Dirichlet series with finite Ritt order on some sequence (x_k), 0 < x_k , as k one has ¦f(x_k)¦=M_t((1 + 0(1) x_k), M_f(x)=sup {¦ f (z) ¦:Re z x}.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 2, pp. 265–269, February, 1991.     

Nonlocal symmetries and the theory of coverings: An addendum to A. M. vinogradov's ‘local symmetries and conservation laws”

For a systemY of partial differential equations, the notion of a covering Y is introduced whereY is infinite prolongation ofY. Then nonlocal symmetries ofY are defined as transformations of which conserve the underlying contact structure. It turns out that generating functions of nonlocal symmetries are integro-differential-type operators.