Even infinite-dimensional real Banach spaces

This article is a continuation of a paper of the first author [V. Ferenczi, Uniqueness of complex structure and real hereditarily indecomposable Banach spaces, Adv. Math. 213 (1) (2007) 462–488] about complex structures on real Banach spaces. We define a notion of even infinite-dimensional real Banach space, and prove that there exist even spaces, including HI or unconditional examples from [V. Ferenczi, Uniqueness of complex structure and real hereditarily indecomposable Banach spaces, Adv. Math. 213 (1) (2007) 462–488] and C(K) examples due to Plebanek [G. Plebanek, A construction of a Banach space C(K) with few operators, Topology Appl. 143 (2004) 217–239]. We extend results of [V. Ferenczi, Uniqueness of complex structure and real hereditarily indecomposable Banach spaces, Adv. Math. 213 (1) (2007) 462–488] relating the set of complex structures up to isomorphism on a real space to a group associated to inessential operators on that space, and give characterizations of even spaces in terms of this group. We also generalize results of [V. Ferenczi, Uniqueness of complex structure and real hereditarily indecomposable Banach spaces, Adv. Math. 213 (1) (2007) 462–488] about totally incomparable complex structures to essentially incomparable complex structures, while showing that the complex version of a space defined by S. Argyros and A. Manoussakis [S. Argyros, A. Manoussakis, An indecomposable and unconditionally saturated Banach space, Studia Math. 159 (1) (2003) 1–32] provides examples of essentially incomparable complex structures which are not totally incomparable.     

Extremely non-complex spaces

We show that there exist infinite-dimensional extremely non-complex Banach spaces, i.e. spaces X such that the norm equality Id+T²=1+T² holds for every bounded linear operator . This answers in the positive Question 4.11 of [V. Kadets, M. Martín, J. Merí, Norm equalities for operators on Banach spaces, Indiana Univ. Math. J. 56 (2007) 2385–2411]. More concretely, we show that this is the case of some C(K) spaces with few operators constructed in [P. Koszmider, Banach spaces of continuous functions with few operators, Math. Ann. 330 (2004) 151–183] and [G. Plebanek, A construction of a Banach space C(K) with few operators, Topology Appl. 143 (2004) 217–239]. We also construct compact spaces K₁ and K₂ such that C(K₁) and C(K₂) are extremely non-complex, C(K₁) contains a complemented copy of C(2^ω) and C(K₂) contains a (1-complemented) isometric copy of ℓ_∞.     

Quantitative Results on Algebraic Independence of Some Numbers with Good Approximations

In this paper, we shall establish transcendence measures of several numbers with good approximations by algebraic numbers. This generalizes the results of Caveny (Proc. Amer. Math. Soc. 120 (1994) 349–357) and Tubbs (Proc. Amer. Math. Soc. 109 (1990) 297–304), and gives a quantitative version of Ably's result (C.R. Acad. Sci. Paris 1(313), série I (1991) 653–655).     

S. G Mikhlin (1908 – 1990)

Our concern is to find a representation theorem for operators in B(c(X), c(Y)) where X and Y are Banach spaces with Y containing an isomorphic copy of c₀. Cass and Gao [1] obtained a representation theorem that always applies if Y does not contain an isomorphic copy of c₀. Maddox [3], Melvin - Melvin [4], and Robinson [5] consider operators in B(c(X), c(Y)) that are given by matrices. In this paper we show that Cass's and Gao's result in [1] can be extended, when Y contains an isomorphic copy of c₀, to certain operators that we call represent able. In addition, we show that when Y contains an isomorphic copy of co there are always operators that fall outside the scope of our representation theorem. Light is also cast on a theorem given in Maddox [3, Theorem 4.2] and [5, Theorem IV].     

Three-dimensional FC Artin Groups are CAT(0)

Building upon earlier work of T. Brady, we construct locally CAT(0) classifying spaces for those Artin groups which are three-dimensional and which satisfy the FC (flag complex) condition. The approach is to verify the ‘link condition’ by applying gluing arguments for CAT(1) spaces and by using the curvature testing techniques of Elder and McCammond [Expositio Math. 11(1) (2002), 143–158].     

Extremal Problems for Operators in Banach Spaces Arising in the Study of Linear Operator Pencils, II

This paper is devoted to the study of operators satisfying the condition

Optimal Domains for Kernel Operators via Interpolation

The problem of finding optimal lattice domains for kernel operators with values in rearrangement invariant spaces on the interval [0,1] is considered. The techniques used are based on interpolation theory and integration with respect to C([0, 1])–valued measures.     

Good Banach spaces for piecewise hyperbolic maps via interpolation

We introduce a weak transversality condition for piecewise C^1+α and piecewise hyperbolic maps which admit a C^1+α stable distribution. We show bounds on the essential spectral radius of the associated transfer operators acting on classical anisotropic Sobolev spaces of Triebel–Lizorkin type which are better than previously known estimates (when our assumption on the stable distribution holds). In many cases, we obtain a spectral gap from which we deduce the existence of finitely many physical measures with basin of total measure. The analysis relies on standard techniques (in particular complex interpolation) but gives a new result on bounded multipliers. Our method applies also to piecewise expanding maps and to Anosov diffeomorphisms, giving a unifying picture of several previous results on a simpler scale of Banach spaces.     

Steenrod Operation in Certain Cobordism Theories

We generalize results of Smirnow [Math. Zam. 65(2) (1999), 270–279] to the cases of MSO and MSU theories. Also we establish relation between the Steenrod operations in MG-cobordism theory (G=O, U, Sp, SO, SU) in our approach and the Steenrod–tom Dieck operations.     

Harmonic analysis in (UMD)-spaces: Applications to the theory of bases

In the paper, a general method for the construction of bases and unconditional finite-dimensional basis decompositions for spaces with the property of unconditional martingale differences is proposed. The construction makes use of a certain strongly continuous representation of Cantor's group in these spaces. The results are applied to vector function spaces and symmetric spaces of measurable operators associated with factors of type II.Translated fromMatematicheskie Zametki, Vol. 58, No. 6, pp. 890–905, December, 1995.     

Linear homeomorphisms of non-classical Hilbert spaces

Let K be a complete infinite rank valued field. In [4] we studied Norm Hilbert Spaces (NHS) over K i.e. K-Banach spaces for which closed subspaces admit projections of norm ≤ 1. In this paper we prove the following striking properties of continuous linear operators on NHS. Surjective endomorphisms are bijective, no NHS is linearly homeomorphic to a proper subspace (Theorem 3.7), each operator can be approximated, uniformly on bounded sets, by finite rank operators (Theorem 3.8). These properties together — in real or complex theory shared only by finite-dimensional spaces — show that NHS are more ‘rigid’ than classical Hilbert spaces.     

On stochastic integral representation of stable processes with sample paths in Banach spaces

Certain path properties of a symmetric α-stable process X(t) = ∫_Sh(t, s) dM(s), t T, are studied in terms of the kernel h. The existence of an appropriate modification of the kernel h enables one to use results from stable measures on Banach spaces in studying X. Bounds for the moments of the norm of sample paths of X are obtained. This yields definite bounds for the moments of a double α-stable integral. Also, necessary and sufficient conditions for the absolute continuity of sample paths of X are given. Along with the above stochastic integral representation of stable processes, the representation of stable random vectors due to[13], Ann. Probab.9, 624–632) is extensively used and the relationship between these two representations is discussed.     

Rough singular integrals on Triebel–Lizorkin space and Besov space

In this paper the authors prove that the homogeneous singular integral T_Ω with ΩH¹(Sⁿ⁻¹) is bounded on the Triebel–Lizorkin spaces and the Besov spaces. These results answer an open problem proposed by Chen and Zhang in [J. Chen, C. Zhang, Boundedness of rough singular integral on the Triebel–Lizorkin spaces, J. Math. Anal. Appl. 337 (2008) 1048–1052]. The same results hold also for the rough singular integral operators T_Ω,h with radial function kernels.     

Application of bases to the study of measures and operators on measure spaces

Summary Under investigation are measures m defined on a σ-algebraA with range in a Banach space X having a Schauder basis {x_n}. Utilizing the corresponding coefficient functionals and coefficient measures, we study the interaction between properties for these measures and properties for the existing measure m. For the spaceCA(A, X) of all countably additive measures of finite variation fromA into X, we show that certain separable subspaces of it are isomorphic-isometric with certain separable subspaces of functions of bounded variation on the interval [0, 1]. For {v_n} inCA(A, X) such that {v_n(A)} converges to v(A) for all A εA a certain ? control measure ? is constructed. An integral for set functions relative to a measure is defined and necessary and sufficient conditions are given for weak convergence. Bounded operators on the space FAC(A,R) of finitelly additive scalar valued set functions onA which are absolutely continuous with respect to λ are considered. Necessary and sufficient conditions for continuity and for compactness are given for Banach space valued operators T defined on such spaces. Weak compactness of T is also studied. Finally, a different kind of representation is given for Banach space valued operators defined on the space FA_v(A, X) of finitely additive set functions fromA into X. Entrata in Redazione il 26 novembre 1976. The first author expresses his gratitude to the Italian Consiglio Nazionale delle Ricerche and the Università degli Studi di Param for the support of his work during his tenure as Visiting Professor of Mathematics. This work was supported by NATO Research Grant No. 835.     

A note on multidimensional modified fractional calculus operators

In the present investigation some new formulas giving the images under multidimensional modified fractional operators of the celebratedH-function of Fox [Trans. Am. Math. Soc. 98 (1961) 395–429] are obtained. Special cases are briefly pointed out and the results are also studied on general spaces of functionsM _γ(R ₊ ⁿ )     

Embeddings in Spaces of Lipschitz Type, Entropy and Approximation Numbers, and Applications

We consider the embeddings of certain Besov and Triebel–Lizorkin spaces in spaces of Lipschitz type. The prototype of such embeddings arises from the result of H. Brézis and S. Wainger (1980, Comm. Partial Differential Equations5, 773–789) about the “almost” Lipschitz continuity of elements of the Sobolev spaces H^1+n/p_p( ⁿ) when 1<p<∞. Two-sided estimates are obtained for the entropy and approximation numbers of a variety of related embeddings. The results are applied to give bounds for the eigenvalues of certain pseudo-differential operators and to provide information about the mapping properties of these operators.     

A linear filtering problem in complete correlated actions

In the last paper, the geometry of the Sz.-Nagy-Foia model for contraction operators on Hilbert spaces was used to advantage in several problems of multivariate analysis. The lifting of intertwining operators, one of the basic results from the Sz.-Nagy-Foia theory, is now recognized as the most adequate operatorial form of the deep classical results of the extrapolation theory. The labeling of the exact intertwining dilations given by [1]Acta Sci. Math. (Szeged) 40 9–32] and the recursive methods used there open a broad perspective for using the Sz.-Nagy-Foia model in multivariate filtering theory. In this paper, using the notion of correlated action (see [5 and 6] Rev. Roumaine Math. Pures Appl. 23, No. 9 1393–1423]) as a time domain, a linear filtering problem is formulated and its solution in terms of the coefficients of the analytic function which factorizes the spectral distribution of the known data and the coefficients of an analytic function which describes the cross correlations is given. In some special cases it is shown that the filter coefficients can be determined using recursive methods from the intertwining dilation theory, of the autocorrelation function of the known data and an intertwining operator, interpreted as the initial estimator given by the prior statistics.     

Inequalities involving upper bounds for certain matrix operators

In this paper, we considered the problem of finding the upper bound Hausdorff matrix operator from sequence spaces l_p(v) (ord(v, p)) intol _p (w) (ord(w, p)). Also we considered the upper bound problem for matrix operators fromd(v, 1) intod(w, 1), and matrix operators frome(w, ∞) intoe(v, ∞), and deduce upper bound for Cesaro, Copson and Hilbert matrix operators, which are recently considered in [5] and [6] and similar to that in [10].     

The diffraction problem for a half plane with different face impedances revisited

The problem of the diffraction of an electromagnetic wave by a half plane with different face impedances is dealt with, following a rigorous approach based on the [L₂⁺( ]₂ theory of systems of Wiener-Hopf equations with piecewise continuous presymbols. The corresponding operator is defined in spaces of physically admissible solutions, the Sobolev spaces H_α⁺( )×H_α−1⁺( ) for , and its Fredholm characteristics are determined. For it is shown that the operators are invertible and their inverses are calculated. In the final section the inverse of a related operator presented by Meister and Speck is also obtained.