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The relative distance between eigenvalues of the compression of a not necessarily semibounded self-adjoint operator to a closed subspace and some of the eigenvalues of the original operator in a gap of the essential spectrum is considered. It is shown that this distance depends on the maximal angles between pairs of associated subspaces. This generalises results by Drmač in [Linear Algebra Appl. 244 (1996), 155-163] from matrices to not necessarily (semi)bounded operators. [ABSTRACT FROM AUTHOR]

We prove observability and null-controllability for quadratic parabolic differential equations. The sensor set is allowed to be sparse and have finite volume if the generator has trivial singular space S. In the case of generators with singular space S ≠ {0} the sensor set is permitted to decay in directions determined by S. The proof is based on dissipation estimates for the quadratic differential operator with respect to spectral projections of partial harmonic oscillators and corresponding uncertainty relations. [ABSTRACT FROM AUTHOR]

We consider the problem of variation of spectral subspaces for bounded linear self-adjoint operators in a Hilbert space. Using metric properties of the set of orthogonal projections as a length space, we obtain a new estimate on the norm of the operator angle associated with two spectral subspaces for isolated parts of the spectrum of the perturbed and unperturbed operators, respectively. In particular, recent results by Kostrykin, Makarov and Motovilov from [Proc. Amer. Math. Soc. 131, 3469-3476] and [Trans. Amer. Math. Soc. 359, 77-89] are strengthened. [ABSTRACT FROM AUTHOR]

An analogue of the Davis-Kahan $${\sin 2\Theta}$$ theorem from (SIAM J Numer Anal 7:1-46, ) is proved under a general spectral separation condition. This extends the generic $${\sin 2\theta}$$ estimates recently shown by Albeverio and Motovilov in (Complex Anal Oper Theory 7:1389-1416, ). The result is applied to the subspace perturbation problem to obtain a bound on the arcsine of the norm of the difference of the spectral projections associated with isolated components of the spectrum of the unperturbed and perturbed operators, respectively. [ABSTRACT FROM AUTHOR]