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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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