ABSTRACT. An analogue of the Davis-Kahan sin 2Θ theorem from [SIAM J. Numer. Anal. 7 (1970), 1-46] is proved under a general spectral separation condition. This extends the generic sin 2θ estimates recently shown by Albeverio and Motovilov in [Complex Anal. Oper. Theory 7 (2013), 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.
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. In this paper we discuss the problem of decomposition for unbounded 2 × 2 operator matrices by a pair of complementary invariant graph subspaces. Under mild additional assumptions, we show that such a pair of subspaces decomposes the operator matrix if and only if its domain is invariant for the angular operators associated with the graphs. As a byproduct of our considerations, we suggest a new block diagonalization procedure that resolves related domain issues. In the case when only a single invariant graph subspace is available, we obtain block triangular representations for the operator matrices.
The problem of variation of spectral subspaces for linear self-adjoint operators under an additive bounded perturbation is considered. The aim is to find the best possible upper bound on the norm of the difference of two spectral projections associated with isolated parts of the spectrum of the perturbed and unperturbed operators.In the approach presented here, a constrained optimization problem on a specific set of parameters is formulated, whose solution yields an estimate on the arcsine of the norm of the difference of the corresponding spectral projections. The problem is solved explicitly. This optimizes the approach by Albeverio and Motovilov in [Complex Anal. Oper. Theory 7 (2013), 1389-1416. In particular, the resulting estimate is stronger than the one obtained there.2010 Mathematics Subject Classification. Primary 47A55; Secondary 47A15, 47B15.
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