The Fuglede theorem and some intertwining relations
- PDF / 318,321 Bytes
- 8 Pages / 439.37 x 666.142 pts Page_size
- 88 Downloads / 261 Views
Advances in Operator Theory https://doi.org/10.1007/s43036-020-00110-5
(0123456789().,-volV)(0123456789().,-volV)
ORIGINAL PAPER
The Fuglede theorem and some intertwining relations Ikram Fatima Zohra Bensaid1,2 • Souheyb Dehimi3 • Bent Fuglede4 Mohammed Hichem Mortad5
•
Received: 20 April 2020 / Accepted: 25 September 2020 Ó Tusi Mathematical Research Group (TMRG) 2020
Abstract In this paper, we show new versions of the Fuglede theorem in an unbounded setting. A related counterexample is also presented. In the second part of the paper, we give a pair of a closed and a self-adjoint (unbounded) operator which is not intertwined by any (bounded or closed) operator except the zero operator. Keywords The Fuglede theorem Intertwining relations Closed and self-adjoint operators
Mathematics Subject Classification 47A05 47B25 47A62
1 Introduction The Fuglede theorem plays a prominent role in the study of normal operators. Probably the main application of this theorem is the fact that it improves the statement of the spectral theorem for normal operators. Recall that Fuglede’s theorem states that if B 2 BðHÞ and A is normal (not necessarily bounded), then Communicated by Elias Katsoulis. & Mohammed Hichem Mortad [email protected]; [email protected] Ikram Fatima Zohra Bensaid [email protected]; [email protected] Souheyb Dehimi [email protected]; [email protected] Bent Fuglede [email protected] Extended author information available on the last page of the article
I. F. Z. Bensaid et al.
BA AB () BA A B: The problem leading to this theorem was first raised by von Neumann in [15] who had already established it in a finite dimensional setting (since this proof is seemingly not well documented, readers may find it in, e.g. Exercise 11.3.29 in [11]). Fuglede was the first one to answer this problem affirmatively in [3] (a quite different proof popped up shortly afterwards and it is due to Halmos [5]). Then Putnam in [17] generalized the result to TA BT () TA B T; where A and B are normal (not necessarily bounded) and T 2 BðHÞ. There is a particular terminology to the transformation which occurs in the Fuglede–Putnam theorem. Definition 1 We say that T 2 BðHÞ intertwines two operators A, B when TA BT. Accordingly, we may restate the Fuglede–Putnam theorem as follows: If an operator intertwines two normal operators, then it intertwines their adjoints. There have been many generalizations of the Fuglede(–Putnam) Theorem since Fuglede’s paper. See, e.g. [9, 13, 18]. For new versions of the Fuglede–Putnam Theorem involving unbounded operators only, readers may wish to consult [8, 10, 16]. Most of these generalizations seem to go into one direction only, that is, towards relaxing the normality hypothesis whilst there are still some unexplored territories as regards the very first version. To get to one main problem of this paper, observe that if A is self-adjoint (and unbounded), then obviously BA AB implies that B A AB for any B 2 BðHÞ. In [6], it was a
Data Loading...
