On determinants of Leżański and Ruston

Determinant Systems in a Certain Class of Algebras

“…If the sequence (D^) is a g-determinant system for an element aeA and oc*0, then the sequence ( aD n ) is also a g-determinant system for a, and the sequence ( We will show that every element aei^ of any g-total algebra A with identity has a g-determinant system but the inverse theorem does not hold. 2) 7 The formula (2.15.1) in the case of linear operators is given by Buraczewski [5] otherwise. We will show that the sequence (S n ) is a g-determinant system for a.…”

Mathematical Analysis

“…His theory of determinants for linear equations in Banach spaces was then expounded by Sikorski [189] in a somewhat modified form corresponding to more symmetric assumptions, by which it would be possible to deduce complete symmetry of the results for a given linear equation and its adjoint. Later Sikorski showed [190] that the nuclear operators always satiszy the postulates of Leza:D.ski, but the converse is not true in general Banach space, although it does hold for a Hilbert space and for reflexive Banach spaces. In the latter cases the approaches of Ruston, Grothendieck and Sikorski coincide [190].…”

“…Later Sikorski showed [190] that the nuclear operators always satiszy the postulates of Leza:D.ski, but the converse is not true in general Banach space, although it does hold for a Hilbert space and for reflexive Banach spaces. In the latter cases the approaches of Ruston, Grothendieck and Sikorski coincide [190]. In his review article on the theory of determinants in a Banach space andin a number of subsequent papers [195,197] Sikorski charac-terizes a class of quasinuclear operators in a Banach space for which it is possible to formulate the Lezaiiski determinant theory.…”