V. A. KHATSKEVICH, V. A. SENDEROV

Copyright © 2013 Sci APM

33

Res.

action of a strict plus-operator A if

AK

Im

.

Proof of Theorem 6. 1) It follows from Theorem 1 that

the factorization (2) and hence the inclusion

12 11

A

2 1

,HH

11

11 12

AA

exist, and hence .

1

11 12

AA

Im

2) By setting , we obtain

1

11 12

QAA

11 12

AA

22 2122 21

BQA AQAA

.

It follows from Lemma 7 that the operator A is

bounded in the norm generated by the decomposition in

Lemma 8. This, as usual, permits considering the linear-

fractional relation A

, where

12

,

HH

is the matrix of the

operator A in the basis . It easily follows from

the inclusion 12

A

that the linear-fractional relation

A

is a closed (multivalued) mapping. Thus, we are

under the conditions of Glicksberg’s theorem [6], which

implies that A

has at least one fixed point in the ball

11X (here and

12

HHX1

is the norm

generated by the new decomposition).

By Lemma 9, this implies that the operator A has a

maximal nonnegative invariant subspace.

3. Conclusions

We note that statement 2) of Theorem 6 holds for any

operator with the property

Im A

P

M

. (3)

Indeed, for any such that , we

have .

Im A

A

In particular, each nonstrict plus-operator in a Krein

space satisfies condition (3).

We note that the method of mapping factorization

constantly finds new applications. In particular, in [18],

precisely this method permits obtaining new conditions

under which the given operator is a plus-operator.

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