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Linear operators sum and Kato-McIntosh’s conjecture. (Sommes d’opérateurs et conjecture de Kato-McIntosh.) (French) Zbl 0951.47001
The paper under review deals with regular (sectorial, closed and densely defined) forms on a fixed complex Hilbert space satisfying Kato’s condition. We recall that a regular form \(\psi\) (or the \(m\)-sectorial operator \(A\) uniquely associated with \(\psi\)) is said to satisfy Kato’s condition if the domain of \(A^{1/2}\) equals the domain of \((A^*)^{1/2}\) and they coincide with the domain of the form \(\psi\).
The author proves that one can associate a maximal accretive operator satisfying Kato’s condition with the sum of two regular forms which satisfy some assumptions concerning the intersection of their domains. As a consequence, if \(A\) and \(B\) are two linear \(m\)-sectorial operators satisfying Kato’s condition then, in some additional hypotheses, there exists a unique \(m\)-sectorial operator \(A\oplus B\) which satisfies the same condition and is the maximal accretive of the algebraic sum \(A+B\). Under certain assumptions on two maximal accretive operators \(A\) and \(B\), more information is given on the numerical range of \(A\oplus B\) in terms of numerical ranges of \(A\) and \(B\); in other words, \(A\oplus B\) satisfies the spectral condition of McIntosh whenever \(A\) and \(B\) do.

MSC:
47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
47A07 Forms (bilinear, sesquilinear, multilinear)
47B44 Linear accretive operators, dissipative operators, etc.
47A12 Numerical range, numerical radius
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