By Radu Gadidov (auth.), Raúl E. Curto, Palle E. T. Jørgensen (eds.)

The conception of operators stands on the intersection of the frontiers of contemporary research and its classical opposite numbers; of algebra and quantum mechanics; of spectral concept and partial differential equations; of the fashionable worldwide method of topology and geometry; of illustration idea and harmonic research; and of dynamical platforms and mathematical physics. the current choice of papers represents contributions to a convention, they usually were conscientiously chosen on the way to bridging diversified yet comparable parts of arithmetic that have just recently displayed an unforeseen community of interconnections, in addition to new and fascinating cross-fertilizations. Our unify ing subject is the algebraic view and method of the learn of operators and their purposes. The complementarity among the range of issues at the one hand and the cohesion of principles at the different has been under pressure. many of the longer contributions symbolize fabric from lectures (in improved shape and with proofs for the main part). in spite of the fact that, the shorter papers, in addition to the longer ones, are a vital part of the image; they've got all been rigorously refereed and revised in order to a cohesion of function, timeliness, clarity, and huge attraction. Raul Curto and Paile E. T.

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**Sample text**

Let A be a Banach algebra and let A, B E A. A multiplication (operator) : A -+ A is defined by the equation MA,B(X) = AXB, X E A. Let A (A 1 ,A2 , ••• ,An), B (BI,B 2 , ••• ,Bn) be n-tuples of elements of A. =l MA;,B;. Clearly, R can have different representations I: MC;,D;. We define the length I of R to be the smallest number of multiplication terms required for any representation of R. Elementary operators have been studied in great detail in recent years. The spectral theory has received much attention [2, 3, 4].

2] R. E. Curto, Spectral theory of elementary operators, preprint. [31 L. A. Fialkow, Spectral properties of elementary operators, Acta. Sci. Math. 46 (1983), 269-282. [4] __, Spectral properties of elementary operators, II, Trans. Amer. Math. Soc. 290 (1985),415-429. [5] __, The range inclusion problem for elementary operators, Michigan Math. J. 34 (1987),451-459. [6] L. A. Fialkow and R. Loebl. Elementary mappings into ideals of operators, Illinois J. Math. 28, 555-578. K. Fong and A. R. Sourour, On the operator identity L: AkX Bk == 0, Can.

Fialkow and R. Loebl. Elementary mappings into ideals of operators, Illinois J. Math. 28, 555-578. K. Fong and A. R. Sourour, On the operator identity L: AkX Bk == 0, Can. J. Math. 31 (1979),845-857. Questions on Bimodules of Nest Algebras Xingde DaP the University of North Carolina at Charlotte, Charlotte, NC 28223 Abstract We will describe some recent results on bimodules of nest algebras and we will propose some related questions. In this notes we will describe some recent results on bimodules of nest algebras.