On a Conjecture of Phadke and Thakare

On a ConJecture of Phadke and Thakare Jochen

Briining

Fachbereich

Mathemutik

der Universitiit

Miinchen

Theresienstrasse 39 8000 Miinchen 2

Federal Republic

of Germuny

and Ki Hang Kim and Fred N. Roush

Mathematics Research Group Alabama State University Montgnne y, Alabama 36101

Submitted by Bichard A. Bmaldi

ABSTRACT We prove the connecteclness of the set of all nonzero bounded linear operators on a complex Hilbert space having a generalized inverse.

In a recent paper [3] S. V. Phadke and N. K. Thakare conjectured that in a complex Hilbert space H the set of operators having a generalized inverse is not connected. The purpose of this note is to disprove this conjecture. We recall that a bounded linear operator A#0 inverse if there is a bounded

on H is said to have a generalized

linear operator

B on H such that

ABA=A. As usual we write ]A( : = (A*A)‘i2 Then

(I)

and denote

(1) is easily seen to be equivalent

by s(]A]) the support of ]A].

to the following

condition:

there is

C > 0 such that

A*A > Cs(lAl). The set of all operators

LINEAR ALGEBRA

with generalized

(2)

inverse will be denoted

by GI(H).

AND ITS APPLZCAZ7ONS 32:113-114(1980)

0 Elsevier North Holland, Inc., 1980

0024-3795/80/040113+2$01.75

113

114

J. BRfJNING, K. H. KIM, AND F. W. ROUSH GI( H) is p&wise

THEOREM.

Proof. inverse,

connected.

Let A # 0 be a bounded

and let

linear operator

on H with generalized

UI A I= A be the polar decomposition tHU((l-

of A. Then

=[0,1],

t)(Al+t~(lAJ)),

is a path in GI(H) in view of (2), connecting A and U. The operators P: = UU* and Q : = U* U are orthogonal projections on H, and we may assume that dim (1, - P)(H) < dim (1, - Q)(H). Now if P is finite, then these dimensions are equal. Consequently, there exists a partial isometry V on H with

W*=l,-P,

connected

V*V=lH-Q.

But

then

with U through a path in GI(H), twu+tv,

U+V

is unitary

and can be

namely

E[O,l].

Next we assume that P is infinite. Then we can find a partial isometry V on H with W* = 1, - P and V* V G 1, - Q. As before, U can be connected with U+ V in GI(H), so we may assume P= 1, from now on. We pick projections P,, Pz on H with P,Pz = 0, P, + Pz = l,, and dim Pl( H) = dimP,( H) = dim H. Then the operators Qi : = U*P, U, i = 1,2, are orthogonal projections, too, satisfying Q1 QZ = 0, Q1 + Q2 = Q, and dim Qj( H) = dim Pi( H) = dim H, i = 1,2. But then also dim (1, - Q,)(H) = dim H, implying that there is a partial define

isometry

W on H with WW* = P2 and W* W= 1, - Q1. We now

u(t):=

uQ1+(1-t)UQ,,

tE[O,l],

uQ,+(t-1)W

tq1,21.

Then U(0) = U, and U(2) is again unitary. Moreover, using (2), it follows that U(t) EGI(H) for t E [0,2]. Since the set of all invertible bounded linear operators on H is connected theorem is proved. We remark

[2, p. 701, U can be connected

that (1) makes sense in an arbitrary

with 1, and the

n W*-algebra.

The above

statement holds also in this more general case; the details of the proof can be found in [l]. REFERENCES 1 2 3

J. Briining, eber Windungszahlen in endlichen W-Algebren und verwandte Fragen, Habilitationsschrift, Marburg, 1977. P. R. HaImos, A Hilbm-t Space Problem Book. Princeton, Van Nostrand, 1967. S. V. Phadke and N. K. Thakare, Generalized inverses and operator equations, Linear Algebra and A$. 23:191-199 (1979).

Received 13 June

1979