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Let $\mathcal{B}(F)$ the algebra of all bounded linear operators on a complex Hilbert space $F$. Let $M\in \mathcal{B}(F)^+$ (i.e. $M^*=M$ and $\langle Mx\;, \;x\rangle\geq 0$ for all $x\in F$).

I want to show that $\mathcal{B}^M(F)\subseteq \mathcal{B}^{M^{1/2}}(F)$, where $$\mathcal{B}^M(F)=\left\{S\in \mathcal{B}(F):\,\,\,\text{Im}(S^{*}M)\subseteq \text{Im}(M)\right\},$$ $$\mathcal{B}^{M^{1/2}}(F)=\left\{S\in \mathcal{B}(F):\,\,\,\text{Im}(S^{*}M^{1/2})\subseteq \text{Im}(M^{1/2})\right\}.$$ Thank you

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The following seems like overkill to me; I'd like to see a solution with less machinery. So I just give a sketch.

  • As $\newcommand{\im}{\operatorname{Im}} \im(M)^\perp = \ker(M)$ we may reduce to the case when $M$ is injective and has dense range, by compressing to $\im(M)$
  • Notice we can work with $S^*$ instead of $S$.
  • So the aim is to show that if $\im(SM) \subseteq \im(M)$ then also $\im(SM^{1/2}) \subseteq \im(M^{1/2})$.
  • As usual, let $M^{-1}$ be the densely defined operator with $D(M^{-1}) = \im(M)$ and $M^{-1}(M\xi) = \xi$.
  • Claim: $\im(SM) \subseteq \im(M)$ if and only if $M^{-1}SM$ is everywhere defined, and bounded (closed graph theorem).

Now consider the strongly continuous one-parameter semigroup $\alpha : t\mapsto M^{it} S M^{-it}$. We analytically extend this, and it turns out that $S$ is in the domain of the $\alpha_i$ if and only if $M^{-1}SM$ is densely defined and bounded, equivalently, $\im(SM) \subseteq \im(M)$. As the domain of $\alpha_i$ is contained in the domain of $\alpha_{i/2}$ this appears to give the claim you want.

Reference: Ioana Ciorănescu and László Zsidó, "Analytic generators for one-parameter groups" see https://projecteuclid.org/euclid.tmj/1178240775 Especially section 6 at the end.

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    $\begingroup$ I think interpolation is the natural framework for this. One has the scale of Hilbert spaces $H_s=D(T^s)$, $T=M^{-1}$, and $S$ maps $H_0\to H_0$, $H_1\to H_1$ by assumption, "so" $S: H_{1/2}\to H_{1/2}$ also, as desired. Of course, that involves roughly the same amount of machinery. $\endgroup$ – Christian Remling Nov 28 '17 at 9:37

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