arXiv subjects
Publications and source records attributed to Dimosthenis Drivaliaris.
We show that a bounded linear operator on a Banach space with closed range has range-kernel complementarity if and only if its generalized amplitude is less than $\pi$. An application to the strong convergence of the iterations of a bounded linear operator is also given.
Let $X$ be a Banach space, $A\in B(X)$ and $M$ be an invariant subspace of $A$. We present an alternative proof that, if the spectrum of the restriction of $A$ to $M$ contains a point that is in any given hole in the spectrum of $A$, then the entire hole is in the spectrum of the restriction.
In this paper, we define the angle of a bounded linear operator $A$ along an unbounded path emanating from the origin and use it to characterize range-kernel complementarity. In particular we show that if $0$ faces the unbounded component of the resolvent set, then $X=R(A)\oplus N(A)$ if and only if $R(A)$ is closed and some angle of $A$ is less than $\pi$.
We show that if the angle of a bounded linear operator on a Banach space, with closed range and closed sum of its range and kernel, is less than $\pi$, then its range and kernel are complementary. In finite dimensions and up to rotations this simple geometric property characterizes operators for which the above complementarity holds. Applying our result we get simple proofs concerning eigenvalues lying in the boundary of the numerical range. Concluding, for an operator on a Hilbert space, we present a sufficient condition, involving the distance of the boundary of the numerical range from the origin, for its range and kernel to be complementary.
In this paper we characterize EP operators through the existence of different types of factorizations. Our results extend to EP operators existing characterizations for EP matrices and give new characterizations both for EP matrices and EP operators.