Segments of bounded linear idempotents on a Hilbert space

Abstract

Let H be a separable Hilbert space. We prove that any two homotopic idempotents in the algebra L(H ) may be connected by a piecewise affine idempotent-valued path consisting of 4 segments at most. Moreover, we show that this constant is optimal provided H has infinite dimension. We also explain how this result is linked to the problem of finding common complements for two closed subspaces of H.

Path connectivity of idempotents on a Hilbert space

“…The following corollary was obtained by J. Giol in [11]. Here, we give an alternative proof, which is based on the ideas and the methods used in Section 2.…”

“…For the sake of convenience, we denote the set of all idempotents in B(H) by P. Two idempotents P and Q in P are said to be homotopic if they can be connected by a continuous path of idempotents in B(H); we shall denote this equivalence relation by P ∼ Q. As is well known, P ∼ Q if and only if dimR(P ) = dimR(Q) and dimN (P ) = dimN (Q) (see [11]), where R(K) and N (K) denote the range and the null-space of an operator K ∈ B(H), respectively.…”

“…Recently, a number of researchers have considered questions concerning the path connectivity between idempotents (see [7,8,9,11]). In 1979, J. Zemánek found that the components of P are arcwise connected ( [12]).…”

“…Particularly, in 2004, J. Esterle had obtained that, for two homotopic idempotents P and Q in a finite dimensional real algebra,s(P, Q) ≤ 3 [9]. In 2005, J. Giol had proved that in an infinite dimensional Hilbert space, for two homotopic idempotents P and Q,s(P, Q) ≤ 4 (see [11]), wheres(P, Q) denotes the minimal number of segments required to connect not only from P to Q, but also from Q to P in P. Moreover, J. Giol had proved in [11] the following result.…”

“…In the present paper, our main task is to improve the above theorem by using the different ideas and methods from that in [11].…”