TAILIEUCHUNG - Báo cáo toán học: "On normal extensions of unbounded operators"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Về phần mở rộng bình thường của các nhà khai thác không bị chặn. | Copyright by INCREST 1985 J. OPERATOR THEORY 14 1985 31-55 ON NORMAL EXTENSIONS OF UNBOUNDED OPERATORS. I J. STOCHEL and F. H. SZAFRANIEC In the theory of unbounded operators an important question is whether the closure of a symmetric operator is selfadjoint or the operator itself is essentially selfadjoint . One of the methods approaching this problem exploits some classes of C -vectors like bounded analytic quasi-analytic ones etc. The canonical theorem in this matter says that a closed symmetric operator is selfadjoint if and only if an appropriate class of its C -vectors is linearly dense in the underlying Hilbert space. This means that any proper candidate for a selfadjoint operator must necessarily have linearly dense sets of each of these classes. However it depends on circumstances which class may be handier to deal with some theorems of general nature like on tensor products can be proved easier using the simplest and the smallest class of bounded vectors 12 whilst in checking whether particular operators for instance differential operators are essentially selfadjoint broader classes of vectors analytic quasi-analytic etc. can serve better. The interplay between symmetric and selfadjoint operators has its counterpart in the case of unbounded normal operators here are formally normal and normal ones. In 12 we have considered this case with the help of bounded vectors emphasizing their relation to bounded operators which fill up the operator in question. The allied problem to essential normality is when a formally normal operator has. a normal extension. Here some phenomenon appears while a symmetric operator always has a selfadjoint extension in some larger space eventually a formally normal need not have any at all 3 Going one step further one can ask about all these unbounded operators which can be extended to normal ones in other words about subnormal operators. In 12 using bounded vectors we have characterized a class of subnormal operators. However .

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