The expression of Moore--Penrose inverse of $A-XY^*$

Abstract: Let

K,, H

be Hilbert spaces and let

L (K, H)

denote the set of all bounded linear operators from

K

to

H

. Let

A i n L (H) r i a n g l e q L (H, H)

with

R (A)

closed and

X, Y i n L (K, H)

with

R (X) s u b s e t e q R (A), R (Y) s u b s e t e q R (A^{*})

. In this short note, we give some new expressions of the Moore--Penrose inverse

(A - X Y^{*})^{+}

of

A - X Y^{*}

under certain suitable conditions.

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