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Linear combinations of right invertible operators in commutative algebras with logarithms. (English) Zbl 0944.47003

In the first part of the paper under review (skipping the preliminaries), one describes a method for solving equations \(Tx=y\), where \(T\) is a linear combination of powers of certain right invertible derivations acting in a commutative algebra \(X\) and \(y\) is an arbitrary element of \(X\). In the author’s words: “The method is, in a sense, a kind of variables separation method.”
In the last part of the paper, one uses a similar method for proving the ill-posedness of certain abstract variants of the homogeneous heat (respectively wave) equation, where the boundary and initial conditions are also homogeneous. One notes that the assumptions here are weaker than the ones appearing in in a previous paper [Math. Nachr. 72, 109-117 (1976; Zbl 0325.34072)].

MSC:

47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
47C05 Linear operators in algebras
47N20 Applications of operator theory to differential and integral equations

Citations:

Zbl 0325.34072
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