Functional Analysis in Interdisciplinary Applications by Tynysbek Sh. Kalmenov Erlan D. Nursultanov Michael V. Ruzhansky & Makhmud A. Sadybekov

Author:Tynysbek Sh. Kalmenov, Erlan D. Nursultanov, Michael V. Ruzhansky & Makhmud A. Sadybekov
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

(2)

where K is an arbitrary bounded linear operator in H with such that . It is also clear that . In particular, is a boundary correct extension of if and only if and .

Lemma 1.

Let be a densely defined correct restriction of the maximal operator in a Hilbert space H. Then if and only if , where L and K are the operators from the representation (1).

Proof.

If then from the representation (1), we easily get

Let us prove the converse. If

then we obtain

(3)

(4)

for all f in H. It follows from (3) that , and from (4) it implies that . Thus . Lemma 1 is proved.

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