Locally defined operators and a partial solution of a conjecture

We prove a weaker form of conjecture concerning the representation theorem for locally defined operators acting between some spaces of differentiable functions presented by K. Lichawski, J. Matkowski and J. Mi? [K. Lichawski, J. Matkowski, J. Mi?, Locally defined operators in the space of differentiable functions, Bull. Polish Acad. Sci. Math. 37 (1989) 315-325].As a corollary we obtain the representation formula for continuous locally defined operators mapping the Banach space C^m(A) into C^k(A), for k≥2 where C^m(A) is the space of m-times Whitney continuously differentiable functions on a perfect set .     

一类广义Lipschitz非线性算子的带误差的Ishikawa迭代程序

借助于周海云和陈东青［４］新近引入的广义Ｌｉｐｓｃｈｉｔｚ概念,研究了实Ｂａｎａｃｈ空间中广义Ｌｉｐｓｃｈｉｔｚ　 －强伪压缩算子的不动点和广义Ｌｉｐｓｃｈｉｔｚ －强增算子方程解的迭代逼近问题,所得结果改进和扩展了近期许多相关的结果,并部分地回答了周海云［３］提出的一个问题．     

Pointwise approximation by Bézier variant of integrated MKZ operators

In this paper the pointwise approximation of Bézier variant of integrated MKZ operators for general bounded functions is studied. Two estimate formulas of this type approximation are obtained. The approximation of functions of bounded variation becomes a special case of the main result of this paper. In the case of functions of bounded variation, Theorem B of the paper corrects the mistake of Theorem 1 of the article [V. Gupta, Degree of approximation to functions of bounded variation by Bézier variant of MKZ operators, J. Math. Anal. Appl. 289 (2004) 292-300].     

Spectral Properties of the Cauchy Operator and Its Product with Bergman's Projection on a Bounded Domain

In this paper exact asymptotic formulae are found for singularvalues of the Cauchy operator and the logarithmic potentialtype operator (on a bounded domain), as well as their productswith Bergman's projection. It is shown that these spectral characteristicsdetect geometric properties of a domain (area and the lengthof the boundary). The hypothesis "can we hear the shape of adrum", from a paper by J.M. Anderson, D. Khavinson, and V. Lomonosov[‘Spectral properties of some integral operators arisingin potential theory’, Quart. J. Math. Oxford (2) 43 (1992)387-407], is correct in the above sense. 1991 Mathematics SubjectClassification: 47B10.     

Maps of several variables of finite total variation. I. Mixed differences and the total variation

Given two points a=(a₁,…,an) and b=(b₁,…,bn) from Rn with a<b componentwise and a map f from the rectangle into a metric semigroup M=(M,d,+), we study properties of the total variation of f on introduced by the first author in [V.V. Chistyakov, A selection principle for mappings of bounded variation of several variables, in: Real Analysis Exchange 27th Summer Symposium, Opava, Czech Republic, 2003, pp. 217-222] such as the additivity, generalized triangle inequality and sequential lower semicontinuity. This extends the classical properties of C. Jordan's total variation (n=1) and the corresponding properties of the total variation in the sense of Hildebrandt [T.H. Hildebrandt, Introduction to the Theory of Integration, Academic Press, 1963] (n=2) and Leonov [A.S. Leonov, On the total variation for functions of several variables and a multidimensional analog of Helly's selection principle, Math. Notes 63 (1998) 61-71] (n∈N) for real-valued functions of n variables.     

A note on compact operators on matrix domains

We establish some identities or estimates for the operator norms and Hausdorff measures of noncompactness of linear operators given by infinite matrices that map the matrix domains of triangles in arbitrary BK spaces with AK, or in the spaces of all convergent or bounded sequences, into the spaces of all null, convergent or bounded sequences, or of all absolutely convergent series. Furthermore, we apply these results to the characterizations of compact operators on the matrix domains of triangles in the classical sequence spaces, and on the sequence spaces studied in [I. Djolovi?, Compact operators on the spaces and , J. Math. Anal. Appl. 318 (2) (2006) 658-666; I. Djolovi?, On the space of bounded Euler difference sequences and some classes of compact operators, Appl. Math. Comput. 182 (2) (2006) 1803-1811].     

Multirectangular characteristic invariants for power ?-Köthe spaces of first type

Let ? be a Banach sequence space with a monotone norm ‖⋅?_‖, in which the canonical system (ei) is a normalized unconditional basis. We consider the problem of quasi-diagonal isomorphism of first type power ?-Köthe spaces E?(λ,a) (see (1) below). From [P.A. Chalov, V.P. Zahariuta, On quasi-diagonal isomorphism of generalized power spaces, in: Linear Topological Spaces and Complex Analysis, vol. 2, METU - TÜB?TAK, Ankara, 1995, pp. 35-44; P.A. Chalov, T. Terzio?lu, V.P. Zahariuta, First type power Köthe spaces and m-rectangular invariants, in: Linear Topological Spaces and Complex Analysis, vol. 3, METU - TÜB?TAK, Ankara, 1997, pp. 30-44; P.A. Chalov, T. Terzio?lu, V.P. Zahariuta, Multirectangular invariants for power Köthe spaces, J. Math. Anal. Appl. 297 (2004) 673-695] it is known that the system of all m-rectangle characteristics μm (see (9) below) is a complete quasi-diagonal invariant on the class of all first type power Köthe spaces [V.P. Zahariuta, On isomorphisms and quasi-equivalence of bases of power Köthe spaces, Soviet Math. Dokl. 16 (1975) 411-414; V.P. Zahariuta, Linear topologic invariants and their applications to isomorphic classification of generalized power spaces, Turkish J. Math. 20 (1996) 237-289], if the relation of equivalency of systems and is defined by some natural estimates with constants independent of m. Deriving the characteristic from the characteristic β (see [V.P. Zahariuta, Linear topological invariants and isomorphisms of spaces of analytic functions, in: Matem. Analiz i ego Pril., vol. 2, Rostov Univ., Rostov-on-Don, 1970, pp. 3-13 (in Russian), in: Matem. Analiz i ego Pril., vol. 3, Rostov Univ., Rostov-on-Don, 1971, pp. 176-180 (in Russian); V.P. Zahariuta, Generalized Mityagin invariants and a continuum of mutually nonisomorphic spaces of analytic functions, Funktsional. Anal. i Prilozhen. 11 (1977) 24-30 (in Russian); V.P. Zahariuta, Compact operators and isomorphisms of Köthe spaces, in: Aktualnye Voprosy Matem. Analiza, vol. 46, Rostov Univ., Rostov-on-Don, 1978, pp. 62-71 (in Russian); P.A. Chalov, P.B. Djakov, V.P. Zahariuta, Compound invariants and embeddings of Cartesian products, Studia Math. 137 (1) (1999) 33-47; P.B. Djakov, M. Yurdakul, V.P. Zahariuta, Isomorphic classification of Cartesian products, Michigan Math. J. 43 (1996) 221-229; V.P. Zahariuta, Linear topologic invariants and their applications to isomorphic classification of generalized power spaces, Turkish J. Math. 20 (1996) 237-289], and using the S. Krein's interpolation method of analytic scale, we are able to generalize some results of [P.A. Chalov, V.P. Zahariuta, On quasi-diagonal isomorphism of generalized power spaces, in: Linear Topological Spaces and Complex Analysis, vol. 2, METU - TÜB?TAK, Ankara, 1995, pp. 35-44; P.A. Chalov, T. Terzio?lu, V.P. Zahariuta, First type power Köthe spaces and m-rectangular invariants, in: Linear Topological Spaces and Complex Analysis, vol. 3, METU - TÜB?TAK, Ankara, 1997, pp. 30-44; P.A. Chalov, T. Terzio?lu, V.P. Zahariuta, Multirectangular invariants for power Köthe spaces, J. Math. Anal. Appl. 297 (2004) 673-695].     

Degree of approximation to function of bounded variation by Bézier variant of MKZ operators

Guo (Approx. Theory Appl. 4 (1988) 9-18) introduced the integral modification of Meyer-Konig and Zeller operators and studied the rate of convergence for functions of bounded variation. In this paper we introduce the Bézier variant of these integrated MKZ operators and study the rate of convergence by means of the decomposition technique of functions of bounded variation together with some results of probability theory and the exact bound of MKZ basis functions. Recently, Zeng (J. Math. Anal. Appl. 219 (1998) 364-376) claimed to improve the results of Guo and Gupta (Approx. Theory Appl. 11 (1995) 106-107), but there is a major mistake in the paper of Zeng. For special case our main theorem gives the correct estimate on the rate of convergence, over the result of Zeng.     

Superposition Operators in the Algebra of Functions of Two Variables with Finite Total Variation

 We show that the Hardy space of functions of two variables with finite total variation is a Banach algebra under the pointwise operations and a suitably chosen norm. Then we characterize Nemytskii superposition operators, which map the Hardy space into itself and satisfy the global Lipschitz condition. Received 7 June 2001; in revised form 8 January 2002     

The Rohde–Schramm theorem via the Gaussian free field

The Rohde–Schramm theorem states that Schramm–Loewner Evolution with parameter κ (or SLE_κ for short) exists as a random curve, almost surely, if κ ≠ 8. Here we give a new and concise proof of the result, based on the Liouville quantum gravity coupling (or reverse coupling) with a Gaussian free field. This transforms the problem of estimating the derivative of the Loewner flow into estimating certain correlated Gaussian free fields. While the correlation between these fields is not easy to understand, a surprisingly simple argument allows us to recover a derivative exponent first obtained by Rohde and Schramm [14], subsequently shown to be optimal by Lawler and Viklund [17], which then implies the Rohde–Schramm theorem.     

Embedding of classes of functions with bounded {\Phi}-variation into generalized Lipschitz spaces

In this note, sufficient and necessary conditions for embedding of classes \({\Phi}\)BV of functions with bounded \({\Phi}\)-variation in Schramm’s sense into generalized Lipschitz classes \({H_{q}^{\omega} (1 \leq q < \infty)}\) are obtained.     

ISHIKAWA TYPE ITERATIVE SEQUENCES WITH ERRORS FOR LIPSCHITZIAN φ-STRONGLY ACCRETIVE OPERATOR EQUATIONS IN ARBITRARY BANACH SPACES

In this paper, we investigate the problem of approximating solutions of the equations of Lipschitzian (?)-strongly accretive operators and fixed points of Lipschitzian (?)-hemicontractive operators by Ishikawa type iterative sequences with errors. Our results unify , improve and extend the results obtained previously by several authors including Li and Liu (Acta Math. Sinica 41 (4)(1998), 845-850), and Osilike (Nonlinear Anal. TMA, 36(1)(1999), 1-9), and also answer completely the. open problems mentioned by Chidume (J. Math. Anal. Appl. 151(2) (1990), 453-461).     

On a problem of Janusz Matkowski and Jacek Wesołowski

We study the problem of the existence of increasing and continuous solutions \(\varphi :[0,1]\rightarrow [0,1]\) such that \(\varphi (0)=0\) and \(\varphi (1)=1\) of the functional equation

$$\begin{aligned} \varphi (x)=\sum _{n=0}^{N}\varphi (f_n(x))-\sum _{n=1}^{N}\varphi (f_n(0)), \end{aligned}$$

where \(N\in {\mathbb {N}}\) and \(f_0,\ldots ,f_N:[0,1]\rightarrow [0,1]\) are strictly increasing contractions satisfying the following condition \(0=f_0(0)<f_0(1)=f_1(0)<\cdots<f_{N-1}(1)=f_N(0)<f_N(1)=1\). In particular, we give an answer to the problem posed in Matkowski (Aequationes Math. 29:210–213, 1985) by Janusz Matkowski concerning a very special case of that equation.

     

The Radon‐Nikodym property,generalized bounded variation and the average range

A generalized bounded variation characterization of Banach spaces possessing the Radon‐Nikodym property is given in terms of the average range. We prove that a Banach space X has the Radon‐Nikodym property if and only if for each function of generalized bounded variation on [0, 1], the average range is a nonempty set at almost all .     

A novel dual approach to nonlinear semigroups of Lipschitz operators

Lipschitzian semigroup refers to a one-parameter semigroup of Lipschitz operators that is strongly continuous in the parameter. It contains -semigroup, nonlinear semigroup of contractions and uniformly -Lipschitzian semigroup as special cases. In this paper, through developing a series of Lipschitz dual notions, we establish an analysis approach to Lipschitzian semigroup. It is mainly proved that a (nonlinear) Lipschitzian semigroup can be isometrically embedded into a certain -semigroup. As application results, two representation formulas of Lipschitzian semigroup are established, and many asymptotic properties of -semigroup are generalized to Lipschitzian semigroup.

     

Asymptotics of large eigenvalues for some discrete unbounded Jacobi matrices

The aim of this paper is to find asymptotic formulas for eigenvalues of self-adjoint discrete operators in given by some infinite symmetric Jacobi matrices. The approach used to calculate an asymptotic behaviour of eigenvalues is based on method of diagonalization, Janas and Naboko’s lemma [J. Janas, S. Naboko, Infinite Jacobi matrices with unbounded entries: asymptotics of eigenvalues and the transformation operator approach, SIAM J. Math. Anal. 36(2) (2004) 643–658] and the Rozenbljum theorem [G.V. Rozenbljum, Near-similarity of operators and the spectral asymptotic behaviour of pseudodifferential operators on the circle, (Russian) Trudy Maskov. Mat. Obshch. 36 (1978) 59–84]. The asymptotic formulas are given with use of eigenvalues and determinants of finite tridiagonal matrices.     

Compactness of operators acting from a Lorentz sequence space to an Orlicz sequence space

LetX andY be closed subspaces of the Lorentz sequence spaced(v, p) and the Orlicz sequence spacel _M , respectively. It is proved that every bounded linear operator fromX toY is compact whenever

Factorization in generalized Calogero–Moser spaces

Using a recent construction of Bezrukavnikov and Etingof, [R. Bezrukavnikov, P. Etingof, Induction and restriction functors for rational Cherednik algebras, arXiv: 0803.3639], we prove that there is a factorization of the Etingof–Ginzburg sheaf on the generalized Calogero–Moser space associated to a complex reflection group. In the case $W=S_n$, this confirms a conjecture of Etingof and Ginzburg, [P. Etingof, V. Ginzburg, Symplectic reflection algebras, Calogero–Moser space, and deformed Harish-Chandra homomorphisms, Invent. Math 147 (2002) 243–348].