On the spectra of sums and the norms of products of orthogonal projections

Viacheslav Rabanovych

Institute of Mathematics (Kiev, Ukraine)

P_{i} is an orthogonal projection on the subspace V_{i} H is a Hilbert space P_{i}^{2} = P_{i} and P_{i}^{∗} = P_{i} Simple properties P_{1} + P_{2} + ⋯ + P_{k} is invertible iff V_{1} + V_{2} + ⋯ + V_{k} = H in the matrix terms

tr (P_{1} + P_{2} + ⋯ + P_{k}) = ∑_{1}^{k} dim V_{i}

∑_{1}^{k} dim V_{i} ≥ rank (P_{1} + P_{2} + ⋯ + P_{k})

Classical result (P. Fillmore, JFA 4, 1969) Theorem 1 An n×n matrix A is a sum of orthogonal projection if and only if

1) A^{∗} = A ≥ 0,

2) tr A ∈ ℤ and

3) tr A ≥ rank A.

Sums of two orthoprojections C. Davis, G. K. Pedersen, P. R. Halmos, P. E. Bjørstad, J. Mandel, M. L. Vasilevski, I. M. Spitkovski, Yu. Samolenko, N. Krupnik, K. Nishio, P. Y. Wu Theorem 2 For non-integer value α, it holds

α ∈ σ(P_{1} + P_{2}) ⟺ 2 − α ∈ σ(P_{1} + P_{2})

Corollary 3 If V_{1} ∩ V_{2} = 0, then the inequality ϵI ≤ P_{1} + P_{2} implies the inequality P_{1} + P_{2} ≤ (2 − ϵ)I. And for complementary projections,

(I − P_{1})(I − P_{2})X_{12}

with ∥X_{12}∥ ≤ 1 − ϵ.

Independent subspaces

Theorem 4 If V_{1}, V_{2}, … V_{k} are linearly independent and P_{1}, P_{2}, … P_{k} are orthoprojections on them, then the inequality

ϵI ≤ P_{1} + P_{2} + ⋯ + P_{k}  (1)

implies the inequality

P_{1} + P_{2} + ⋯ + P_{k} ≤ (k − (k − 1)ϵ)I

and the inequality

∥(I − P_{1})(I − P_{2}) … (I − P_{k})∥ ≤ 1 − ϵ

The last inequality if far advanced hypothesis and it is true up to now only for k = 2.

PROOF. From (1) using the perturbation theory for sums of positive operators, we can obtain that

ϵP_{Im P_{1}+···+Im P_{k−1}} ≤ P_{1} + ⋯ + P_{k−1}

and as a corollary

ϵP_{Im P_{i}+Im P_{j}} ≤ P_{i} + P_{j}.

This implies

∥P_{i}P_{j}∥ ≤ 1 − ϵ

and

∥A − I∥ ≤ (m − 1)(1 − ϵ)

Pairwise orthogonality Suppose in addition V_{i} ⊥ V_{j} when j ∉ N_{i} and for all i, |N_{i}| ≤ m. Theorem 5 The inequality

ϵI ≤ P_{1} + P_{2} + ⋯ + P_{k}

implies the inequality P_{1} + P_{2} + ⋯ + P_{k} ≤ (m − (m − 1)ϵ)I.

Dependent subspaces There exit orthoprojections such that ∥ ∑_{1}^{k} P_{i}∥ is close to m (to k in the case without orthogonality). Really an m − 1 × m − 1 matrix

A = [\begin{matrix}m − 1 − (m − 2)β & 0 & … & 0 & 0 \\ 0 & β & … & 0 & 0 \\ ⋮ & ⋮ & ... & ⋮ & ⋮ \\ 0 & 0 & … & β & 0 \\ 0 & 0 & … & 0 & β \end{matrix}]

is a sum of m − 1 orthoprojections P_{1}, …, P_{m−1} and if P_{m} = diag (0, 1, …, 1) then P_{1}+P_{2}+⋯+P_{m} = diag (m−1−(m−2)β, 1+β, …, 1+β) Hypothesis 6 The norm of the following product depends on m

∥(I − P_{1})(I − P_{2}) … (I − P_{k})∥ ≤ 1 − \frac{const}{m}

Although there exit orthoprojections such that

∥(I − P̂_{1})(I − P̂_{2}) … (I − P̂_{k})∥ ≥ 1 − \frac{Const}{m}

Inexact local solves Assume that we have new Hermitian operators P̃_{i} with

Im P̃_{i} = Im P_{i} and ∥P_{i} − P̃_{i}∥ ≤ δ

for all i, and some δ > 0. Then we can formulate a theorem similar to Theorem 5 Theorem 7 The inequality

ϵI ≤ P̃_{1} + P̃_{2} + ⋯ + P̃_{k}

implies the inequality P̃_{1} + P̃_{2} + ⋯ + P̃_{k} ≤ (1 + (1 + δ)(m − 1)(1 − ϵ))I