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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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