Idempotents, their HERMITian Components, and Subspaces in Position p of a HILBERT Space

Abstract

Introduction.Let H be a complex HILBERT space and let L(H) be the algebra of all bounded linear operators on H . Write I for the identity and 0 for the zero operator of L(H). Use similar notations with respect to other spaces.An idempotent SEL(H) is an operator which satisfies the equation S l = S . Assume, in addition, that S is non-trivial, i.e. that S i O , I , and that S is non-HERMITian if nothing else is specified. Then H has the decomposition H = kerS i r a n S , where ranS=ranS. \Vrite rpS for the range projection of S , and denote ReS and ImS its HERMITian components: ReS= ~ (8+8*), I m 8 = ~. (8-8*). The HERMITian components lead to the following CARTESian decomposition of S : S=ReS+iImS. -If S is an idempotent, then (1-5) is an idempotent, the complementary idempotent of S . Moreover, the adjoints S* and ( I -5)* are idempotents too. The involution corresponding to S is (2s -1). One has ( 2 5 -I)2 = I . If s is HERMITian, i.e. a projection, then (25-1) is a symmetry. Let BCL(H) be an arbitrary operator. Let a(B) denote the spectrum of B. I n particular, write up for the point spectrum, u, for the continuous spectrum and zz for the approximate point spectrum. Let r(B) denote the spectral radius and let j IBl 1 denote the norm of B : llB[I=H~ip {IlBfIl 1 ilfil= 1, f C H } . Moreover, let IBl denote the root (B*B)'12 and let B+ = ~ (/BI + B ) , B-= -( j B [ -B ) be the definite components of B. Observe also the decompositions H = kerB@ranB* = kerB* @ @ r a i i . Assume now that B is invertible and denote, in addition, by cond(B) the condition number of B : cond(B) =IIB[I ~~B -~~~, where IIB-lll=(inf {I[Bf[l I llf[l= 1, f c H } ) -* . Further, if B is Hermitian denote its lower bound m(B), where m(B) = =inf { ( B f , f ) I ]lfil= 1 , f C H } and its upper bound M ( B ) , where M ( B ) =sup {(Bf, f )

Geometry of unitary orbits of oblique projections

“…is the unique sefadjoint projection onto S. Note that, by this formula, p ∈ A when q ∈ A. Several different formulas are known for p (see [11], p. 294); perhaps the simplest one is the so-called Kerzman-Stein formula…”

“…The space Q of all idempotents of a C * -algebra (or, more generally, of a Banach algebra) has a rich topological and geometrical structure, studied for example in [17], [29], [11], [24], [7] and [8].…”

“…This paper, which started from a close examination of Pasternak-Winiarski's work, is part of the program of understanding the structure of the space of idempotent operators. For a sample of the vast bibliography on the subject the reader is referred to the papers by Afriat [1], Kovarik [15], Zemánek [29], Porta-Recht [24], Gerisch [11], Corach [6] and the references therein. Applications of oblique projections to complex, harmonic and functional analysis and statistics can be found in the papers by Kerzman and Stein [13], [14], Ptak [27], Coifman and Murray [5] and Mizel and Rao [17], among others.…”

Spectrum of the Kerzman?Stein Operator for Model Domains

“…Moreover, there is a general relationship between the norm of a projection operator and the norm of its skew-hermitian part. This is described by Gerisch in [5], for instance.…”