Anderson, William N.

Schreiber, Morris

English

Anderson William N.

Schreiber Morris

University of Szeged

The infimum of two projections By W. N. ANDERSON, Jr. and M. SCHREIBER in New York (N.Y., U.S.A.) 1. The purpose of this note is to give a closed formula for the infimum of two projections in Hilbert space. Heretofore, this infimum has been expressed by an iterative scheme as follows:1) If P and Q are projections on closed subspaces M and N, then the projection PA Q on M D N is given by (1) P A ( 2 = l i m P{QPf. We shall show that (2) PAQ = 2P(.P+Q)*Q if M+N is closed, where (P+Q)+ denotes the inverse of P+Q restricted to its range. This formula is exactly that given by A N D E R S O N and D U F F I N [2] for matrices, and the result of this note is that it generalizes to infinite dimensions as stated. We also give an apparently heretofore unstated necessary and sufficient condition for M+N to be closed (namely, that M and TVmake a positive angle modulo M(~)N). The combination A(A+B)+B, introduced in [2], is called the parallel sum, so named because of its origin in and application to electrical network theory ( ( T 1 + r 2 O " 1 = r i ( r i + r i ) ~ l r2 is the resistance arising f rom resistors r t and r 2 in parallel). A second formulation of the parallel sum in finite dimensions is given b y A N D E R S O N [1] . We shall use the following notations: PM for the projection on M, R(A) and N(A) for range and nullity, M~ and ML for closure and orthocomplement, A\M for restriction. 2. We begin by noting some facts about non-negative bounded operators. For hermitian operators generally one knows that R(A)—AR{A), that A is one-one on R{A), and A is invertible on R(A) if and only if i?(^4)=i?(/4) (closed graph theorem). ') See [4] for background. 166 W. N. Anderson—M. Schreiber Hence for non-negative operators we have (i) R(A) closed if and only if A is bounded away f rom 0 on -R(/l). It follows that R(A) is closed if and only if R(A*) is closed. Since N(A) = N(A*) (because by positivity (Ax, x)=0 if and only if x = 0), and therefore R(A) = R(A*), we have (ii) R(A) closed implies R(A) = R(A>). T . CRIMMINS has shown that R(S) + R(T) = (SS* + TT*)± for any bounded operators S and T.2) From this we get (iii) If .KOI), R(B), and R(A +B) are closed, then (3) ' + for R(A) + R(B) = R(A*)+R(Bi) = R((A+B)i) = R(A+B). As it follows f rom the preceding computation that /?(A) + /?(fi) is closed if and only if R((A +B)i) is closed, we have (iv) If and R(B) are closed, a necessary and sufficient condition that R(A +B) be closed is that / ? ( / ! ) ( 5 ) be closed. Next, we introduce a generalized inverse (see [3] for background) and the parallel sum. If A has closed range; define A+ by (4) A + \R = 0 4 | R ) - \ ^ + If A+B has closed range, define the parallel sum A:B by (5) A:B = A(A + B)+ B. This operation satisfies -(6) A-.B = B-.A =.(A:B)*,: rR(A:B) = R(A)DR(B) if R(A) and R(B) are closed, as we shall presently sh6w. The equations R(A+B) = = R(A) + R(B) and R(A\B) = R(A) Pi R(B), valid when R(A) + R(B) is closed, show, that the set of A £ 0 with .closed range is almost a lattice, in the sense that the sup and inf, inherited f rom the lattice of.subspaces, exist whenever the sum of the ranges is closed. If^we now think.of projections P and Q in this context and ask what significance attaches to their sup and inf, we are led to the following theorem: 2) The result is unpublished and was told to us by J. P. WILLIAMS. Proof: Write A = ( o ~ o ) o n / / f f i / / - Then {/?0S) + R(T)}©{0} = R(A) = = R((SS* + ft*))*'® {0}. ' The infimum of two projections 167 T h e o r e m . If P and Q are projections such that R(P) + R(Q) is dosed, then 2P:Q = PhQ. Note that by (iv) the hypothesis is equivalent to assuming R(P + Q) closed. To prove the theorem, we first prove the identities (6) (following [2]). By (iii) we have/?04), R(B) c R(A + B). Hence B(A+B)+(A + B) = B,(A+B)+(A + B)A = A (because (A +B)+(A+B) is the projection on R(A +B)). Then A:B = A(A+B)+B = ' = (A+B-B)(A+B)+(A+B-A) easily reduces to B(A+B)+A = B:A. Since (A:B)* = B.A, we have (A:B)* = A:B. By the commutativity again we have R(A:B) c R ( A ) D R ( B ) . If x 6 « ( / 1 ) 0 t h e n • . (A:B)(A + + B + ) x = A(A + B)+BB + x + B(A + B) + AA + x = . = ( A ( A + B ) + + B ( A + B ) + ) x = PRU+B)x = x. •Hence R(A:B) = R(A) f]R(B). This last computation simplifies, if A and B are pro-jections P and Q, to 2 P ( P + Q)^Qx = 2 ( P : Q ) x = x for x^R(P) DR(Q). Thus 2(P-Q) is a hermitian idempotent with range- R(P) f l R(Q), and this proves the theorem, . - . 3. We conclude by showing that R(P) + R(Q) is closed if and only if R(P) and R(Q) make a positive angle modulo R(P)n R(Q). To establish this, we first consider the case when the two subspaces, which we denote by M and N, are dis-joint. Then the assertion is t h a t ' M + N is closed if and only if (7) . \(x, y)\ M M l ( 1 - 5 ) > 0 ) for all x £ M, y £ Â . Suppose (7) holds, and consider a Cauchy sequence un = x„+yn in M+N. Write unm = un — um, etc. Then Ik, J 2 - ll-̂ nmli2 + Ibnmll2 + 2 R e ( x n m , y „ m ) ^ ||x„m|[2 +1|J|2-- 2 ( l . - ^ ) | | x „ J M b „ J | = { | | x n J | - | b n J ! } 2 + 2 5 | | x „ J | | b „ J . It follows that u„m —0 implies xnm -^0, ynm - 0 , and therefore that u„ converges in M+N.3) Conversely, suppose M+N is closed. It then follows that the "coordinate" map T: M+N M given by T{x+y) — x is continuous. The proof consists in verifying that T has closed graph and then applying the closed graph theorem. This argument is due to KOBER [5], and as the verification is simple we omit it. Since T is bounded, we have 3) This formulation of the argument is due to B . GLICKFELD. 168 W. N. Anderson—M. Schreiber: The infimum of two projections f o r a l l x£M, y£N. T a k i n g [|x|| = ||y\\ = 1 a n d s q u a r i n g , w e ge t 1 A2(2±2Re(x,y)) w h e n c e | R e ( x , ^ 1 — ^ r W , a n d (7) f o l l o w s f r o m t h i s i m m e d i a t e l y . Z.A I f MC\N = I ^ {0}, w e p a s s t o t h e q u o t i e n t s p a c e HI I. I f M+N i s c l o s e d , s o is (M+N)/I — M/I+NjI a n d so M a n d N m a k e a p o s i t i v e a n g l e m o d u l o I. C o n -verse ly , if Mjl a n d N/I m a k e a p o s i t i v e a n g l e HI I, t h e n M/I+N/I is c l o s e d , w h e n c e s o is M + N . References [1] W . N. ANDERSON, Jr., Shorted Operators, to appear in SIAM J. Applied Math. [2 ] W . N . ANDERSON, Jr. and R . J. D U F F I N , Series and Parallel Addition of Matrices, J. Math. Anal. Appl., 26 (1969), 576—594. [3] A. BEN ISREAL and A. CHARNES, Contributions to the Theory of Generalized Inverses, J. Indust. Appl. Math., 11 (1963), 667—699. [ 4 ] P. R . HALMOS, A Hilbert Space Problem Book, Van Nostrand (Princeton, 1 9 6 7 ) . [5] H. KOBER, A Theorem on Banach Space, Comp. Math.,1 (1939), 135—140. ROCKEFELLER UNIVERSITY NEW YORK, NEW YORK 1 0 0 2 1 (Received January 15, 1971) 

The infimum of two projections

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The infimum of two projections

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