TAILIEUCHUNG - Báo cáo toán học: "The Birman-KreÄ­n-Vishik theory of self-adjoint extensions of semibounded operators "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Các lý thuyết Birman-KreÄn-Vishik tự liên hợp phần mở rộng của các nhà khai thác semibounded. | Copyright by INCREST 1980 J. OPERATOR THEORY 4 1980 251-270 THE BĨRMAN-KREỈN-VISHIK THEORY OF SELF-ADJOINT EXTENSIONS OF SEMIBOUNDED OPERATORS ALBERTO ALONSO and BARRY SIMON 1. INTRODUCTION The von Neumann theory 6 of self-adjoint extensions of a closed symmetric operator A on a Hilbert space is given in many textbook presentations. It sets up a one to one correspondence between self-adjoint extensions A U and uni-taries u from Ker 4 i to Ker A i .This parametrization is continuous in a suitable sense see 4 . The von Neumann theory is so elegant and complete that it appears to close the subject of the general theory but this is definitely not the case. with respect to extensions of zfs which are bounded from below. Again from the textbooks one learns about the Friedrichs extension 2 but its relation to the von Neumann theory is not clear. Ocasionally mention is made of the theorem of Krein 5 that if A has finite deficiency indices and is bounded from below then all self-adjoint extensions of A are also bounded from below. There is a much deeper and most natural analysis possible for the self-adjoint extensions of a semibounded operator. This analysis was presented in part in the above paper of Krein and completed in papers of Birman 1 and Vishik 12 . One purpose of the present paper is to make propaganda for this Birman-Krein-Vishik theory which seems to be virtually unknown in the English language literature In this sense the present paper is semi-expository. There is an important difference of emphasis in our presentation. Much of the original spectral theory of bounded self-adjoint operators was discussed in terms of quadratic forms. The development of the theory of unbounded operators by von Neumann deemphasized forms and exploited heavily operator ideas especially graphs and Cayley transforms. The papers of Kreln-Birman-Vishik extensively use such ideas and base their analysis on them. They are somewhat unusual for their period in that they do consider what .

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