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For an unbounded operator $S$ on a Banach space the existence of invariant subspaces corresponding to its spectrum in the left and right half-plane is proved. The general assumption on $S$ is the uniform boundedness of the resolvent along the imaginary axis. The projections associated with the invariant subspaces are bounded if $S$ is strictly dichotomous, but may be unbounded in general. Explicit formulas for these projections in terms of resolvent integrals are derived and used to obtain perturbation theorems for dichotomy. All results apply, with certain simplifications, to bisectorial operators.
Let $mathscr{H}^2$ denote the Hilbert space of Dirichlet series with square-summable coefficients. We study composition operators $mathscr{C}_varphi$ on $mathscr{H}^2$ which are generated by symbols of the form $varphi(s) = c_0s + sum_{ngeq1} c_n n^{
Let $X$ be a space of homogeneous type and let $L$ be a nonnegative self-adjoint operator on $L^2(X)$ which satisfies a Gaussian estimate on its heat kernel. In this paper we prove a Homander type spectral multiplier theorem for $L$ on the Besov and
Let $sigma(A)$, $rho(A)$ and $r(A)$ denote the spectrum, spectral radius and numerical radius of a bounded linear operator $A$ on a Hilbert space $H$, respectively. We show that a linear operator $A$ satisfying $$rho(AB)le r(A)r(B) quadtext{ for all
For a pointwise multiplier $varphi$ of the Hardy-Sobolev space $H^2_beta$ on the open unit ball $bn$ in $cn$, we study spectral properties of the multiplication operator $M_varphi: H^2_betato H^2_beta$. In particular, we compute the spectrum and esse
An arbitrary linear relation (multivalued operator) acting from one Hilbert space to another Hilbert space is shown to be the sum of a closable operator and a singular relation whose closure is the Cartesian product of closed subspaces. This decompos