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Algebraic sum of unbounded normal operators and the square root problem of Kato. (English) Zbl 1121.47015
Taking \(A\) and \(B\) to be unbounded normal operators on a complex Hilbert space \({\mathbb H}\), we may, as a consequence of the spectral theorem, write \(A=A_1-iA_2\) and \(B=B_1-iB_2\), where the \(A_k\) and \(B_k\) are unbounded selfadjoint operators on \({\mathbb H}\). If, moreover, the operators \(A_k\) and \(B_k\) are non-negative, then, considering the sesquilinear functionals defined by \(\phi(u,\,v):=\langle A_1u,\,v\rangle - i\,\langle A_2u,\,v\rangle\), \(\psi(u,\,v):=\langle B_1u,\,v\rangle - i\,\langle B_2u,\,v\rangle\), and \(\xi(u,\,v):=\phi(u,\,v)+\psi(u,\,v)\), we see that if \(\phi\) and \(\psi\) are sectorial, that is, if there exist constants \(c_1\) and \(c_2\)with \(\operatorname{Im} \phi(u,\,u)\leq c_1\operatorname{Re}\phi(u,\,u)\) and \(\operatorname{Im} \psi(u,\,u)\leq c_1\operatorname{Re}\psi(u,\,u)\) for all \(u\) in the appropriate domains, then \(\xi\) is sectorial as well.
Theorem 2.1 of the present paper demonstrates that, under the further assumptions that the intersection of the domains of \(A\) and \(B\) is dense in \({\mathbb H}\) and that the operator \(\overline{A+B}\) is maximal, then this latter operator satisfies the square root problem of Kato; that is, the domains of \(\overline{A+B}^{1/2}\) and \(\overline{A+B}^{\,*1/2}\) both coincide with the intersection of the domains of \(A^{1/2}\) and \(B^{1/2}\). The density assumption on \(\text{Dom}(A)\cap\text{Dom}(B)\) can be replaced with certain conditions on \(\text{Dom}(|A|^{1/2})\cap\text{Dom}(|B|^{1/2})\) that ensure (Theorem 2.2) the existence of an operator \(A\oplus B\) (a “generalized” sum of \(A\) and \(B\)) satisfying the square root problem of Kato. (So \(\text{Dom}((A\oplus B)^{1/2})=\text{Dom}((A\oplus B)^{\,*1/2})=\text{Dom}(|A|^{1/2})\cap\text{Dom}(|B|^{1/2})\).)
The paper concludes with an example where the sum \(A+B\) is a perturbed Schrödinger operator.
MSC:
47B25 Linear symmetric and selfadjoint operators (unbounded)
35J15 Second-order elliptic equations
47D08 Schrödinger and Feynman-Kac semigroups
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References:
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