Subformula property for some modal logics, such as the smallest normal logic K with the extra axiom $A\supset \square \square A,\square ◊A\supset A,◊\square A\supset \square \square A$, or the like, is studied by devising appropriate sequent calculi.

A new version of analytically saturated sequents, the original of which has been introduced in (Rep Math Logic 53: 43–65, 2018), plays the principal role throughout this paper.

A bounded linear operator $T$ on a Hilbert space $\mathcal{H}$ is called to be hyponormal if and only if $T^{*}T\ge T{T}^{*}$. We study the operator $C:=T^{*-1}T$ for an invertible hyponormal operator $T$ and show that (i) $C$ is doubly power bounded, that is ${\sup }_{n\in \mathrm{\mathbb{Z}}}\Vert C^n\Vert <\infty$, so it is similar to unitary (on a Hilbert space $\mathcal{K}$) [5] and hence $C$ is a spectral operator of scalar type. (ii) For each disjoint Borel sets ${\delta }_1,\cdots ,{\delta }_n\subset S^1:=\{z\in \mathrm{ℂ}:|z|=1\}$ such as ${\delta }_1\cup \cdots \cup {\delta }_n=S^1, T$ can be decomposed to the sum of hyponormal operators $TF\left({\delta }_j\right)$ where $\left\{F\right(·\left)\right\}$ is a spectral measure of $C$ with $C={\int }_{[0,2\pi )}{e}^{i\theta }d{F}_{\theta }$. In particular, if (positive, invertible) then $C$ is similar to a unitary operator on the same Hilbert space $\mathcal{H}$. (iii) If $F\left(\delta \right)$ is self-adjoint, i.e., it is an orthogonal projection, then $F\left(\delta \right)\mathcal{H}$ is a reducing subspace of $T$. (iv) If $\lambda \in \sigma \left(C\right)$ is an isolated point of $\sigma \left(C\right)$ then $\lambda$ is an eigenvalue of $C$ and $F\left(\right\{\lambda \left\}\right)$ is self-adjoint with $F\left(\right\{\lambda \left\}\right)\mathcal{H}=\ker (C-\lambda )=\ker (C-\lambda {)}^{*}$. (v) An inequality of Putnam type for $TF\left(\delta \right)$ and (vi) If both $R=\mathfrak{R}\mathfrak{e}T$ and $S=\mathfrak{I}\mathfrak{m}T$ are positive invertible then $R^{-1/2}T{R}^{-1/2}$ has a spectral decomposition

where $U$ is unitary similar to $C$ and $U={\int }_{e^{i\theta }\in \sigma \left(U\right)}{e}^{i\theta } d{E}_{\theta }$ is its spectral decomposition.