The Angle Between Subspaces of a Hilbert Space

  title={The Angle Between Subspaces of a Hilbert Space},
  author={Frank Deutsch},
This is a mainly expository paper concerning two different definitions of the angle between a pair of subspaces of a Hilbert space, certain basic results which hold for these angles, and a few of the many applications of these notions. The latter include the rate of convergence of the method of cyclic projections, existence and uniqueness of abstract splines, and the product of operators with closed range. 
The rate of convergence in the method of alternating projections
A generalization of the cosine of the Friedrichs angle between two subspaces to a parameter associated to several closed subspaces of a Hilbert space is given. This parameter is used to analyze the
When is the sum of closed subspaces of a Hilbert space closed
We provide a sufficient condition for a finite number of closed subspaces of a Hilbert space to be linearly independent and their sum to be closed. Under this condition a formula for the orthogonal
A classification of projectors
A positive operator A and a closed subspace S of a Hilbert space H are called compatible if there exists a projector Q onto S such that AQ = Q∗A. Compatibility is shown to depend on the existence of
On the angle and the minimal angle between subspaces
We study those pairs of subspaces of a complex Hilbert space with the same angle and minimal angle and present several characterizations of the pairs of subspaces with angle equal to .
Projections in operator ranges
If H is a Hilbert space, A is a positive bounded linear operator on H and S is a closed subspace of H, the relative position between S and A -1 (S⊥) establishes a notion of compatibility. We show
On the Reduced Minimum Modulus of Projections and the Angle between Two Subspaces
Let M and N be nonzero subspaces of a Hilbert space H, and PM and PN denote the orthogonal projections on M and N , respectively. In this note, an exact representation of the angle and the minimum


On the convergence of von Neumann's alternating projection algorithm for two sets
We give several unifying results, interpretations, and examples regarding the convergence of the von Neumann alternating projection algorithm for two arbitrary closed convex nonempty subsets of a
On the method of cyclic projections for convex sets in Hilbert space
The method of cyclic projections is a powerful tool for solving convex feasibility problems in Hilbert space. Although in many applications, in particular in the field of image reconstruction
On a calculus of operators in reflexive vector spaces
Linear transformation theory in general vector spaces is not nearly as extensive as it is for that special space, Hubert space. In Hubert space large and important classes of transformations, the
Error bounds for the method of alternating projections
The method of alternating projections produces a sequence which converges to the orthogonal projection onto the intersection of the subspaces, and the sharpest known upper bound for more than two subspaced is obtained.
Introduction to Hilbert spaces with applications
Normed Vector Spaces The Lebesgue Integral Hilbert Spaces and Orthonormal Systems Linear Operators on Hilbert Spaces Applications: Applications to Integral and Differential Equations Generalized
Convergence of abstract splines
Rate of Convergence of the Method of Alternating Projections
A proof is given of a rate of convergence theorem for the method of alternating projections. The theorem had been announced earlier in [8] without proof.
Theory of Reproducing Kernels.
Abstract : The present paper may be considered as a sequel to our previous paper in the Proceedings of the Cambridge Philosophical Society, Theorie generale de noyaux reproduisants-Premiere partie
On certain inequalities and characteristic value problems for analytic functions and for functions of two variables
the constant r = (1 +6)/(1 -6) being greater than 1. * Presented to the Society, October 31, 1936; received by the editors December 6, 1935, and January 27, 1936. t The same space was investigated in