ON THE PRODUCT OF SELF-ADJOINT OPERATORS

A proof is given for the fact that the product of two self-adjoint operators, one of which is also positive, is again self-adjoint if and only if the product is normal. This theorem applies, in particular, if one operator is an orthogonal projection. In general, the positivity requirement cannot be dropped.


INTRODUCT ION.
Products of self-adjoint operators in Hilbert space play a role in several dif- ferent areas of pure and applied mathematics.We shall give three examples: a.In the simplified Hilbert space model of quantum mechanical systems, measur- able quantities a,b,... (location, momentum, etc.) are represented by self-adjoint operators ("observables") A,B,... (Mackey [1,2]).The state of the system itself is given by the so-called "statistical operator".W, which is positive with trace (W) 1 and also named the "density operator" of the system.This probabilistic parlance stems from the intrinsic stochastic nature of quantum mechanics: property a, say, with representing operator A, will be found in the system not with certainty, but with a probability given by PW (a) trace (WA), and by measuring a, the original system changes into a new one whose density or state is given by AWA trace (WA) W. REHDER (see Lders [3], and for a recent discussion, Bub [4]).W' determines the conditional probability of "b given a" via Pw(bla) trace (W'B).
If A and W commute, A is called "objective" with respect to W, and WA is a new observ- able of the system.If A and B commute, AB represents the property "a and b"-.
b. Every bounded operator T may be written T A + iB with A and B self-adjolnt.
If T is already known to be seml-normal, Putnam [5, p. 57] proved that normality and self-adjointedness of AB are the same.c.Radjavi and Rosenthal [6] proved that the product of a positive and a self- adjoint operator always has a non-trivlal invariant subspace.It has not yet been decided whether the product of two self-adjoint operators or, more generally, of a positive and a unitary operator has an invarlant subspace (this is the famous "invarlant subspace problem").
The starting point for the discussion in the present note is the following theo- rem (all operators are supposed bounded).

MAIN RESULTS.
THEOREM.Let A and B be self-adjoint, and A or B be positive.Then AB is self- adjoint if and only if AB is normal.
PROOF.Of course the "only if" implication is obvious.As to the converse, we use the well-known Fuglede-Putnam theorem [7,8]  To prove our result, set N 1 BA and N 2 AB N 1 A 2 i.e commutes with B. Since A is positive, A is the square root of A 2 and hence A commutes with B. (If B is positive, exchange the roles of A and B).
This theorem characterizes the self-adjoint operators in the class of normal operators; it is known that every self-adjoint operator T can be written in its polar decomposition as a product T AB with A positive and B unitary.Here B is even self- PROOF.M reduces B iff AB BA, i.e., AB is self-adjolnt.Since A is positive, our theorem applies.
COROLLARY 3. Let A and B be orthogonal projections.Then the comutatlvlty re- lation AB BA is equivalent to ABA BAB.
PROOF.AB BA means that AB is self-adjolnt, whereas ABA BAB expresses nor- mality of AB.
The fact that ABA BAB implies AB BA may also be seen directly by evaluating (ABA AB)*(ABA AB) 0.
We give now an example showing that the positivity requirement in the theorem cannot be dropped: the self-adjolnt matrices A fulfill AB -BA, so that AB is normal but not self-adjoint.The reason, according to our theorem, is that neither A nor B are positive.From this we conclude the follow- ing weakening of the theorem; COROLLARY 4. Let A and B be self-adjoint, and A or B be positive.If AB BA#0, then also AB + BA # 0. (If the commutator of A and B is non-zero, then their anti- commutator is also non-zero).
On the other hand, the assumptions of the theorem are not necessary: If B and AB are self-adjoint, it is not necessary that A even be normal: take B as above and What about the other partial converse of the theorem?If A is positive and AB self- which states the following: For nor- (Rudin [9, p.315] Proof (b) of Theorem 12.35).Our theorem states the converse: all operators T AB with A positive and B self-adjolnt are al- PRODUCT OF SELF-ADJOINT OPERATORS 815 adjoint, because T is