A Note on Square Roots of Positive Operators

Abstract

1. The study of the self-adjoint operators of mathematical physics by variational methods (as, for instance, in the monograph of Mikhlin [1]) utilizes Hilbert space theory in an elementary sense, in that no use is made of the spectral theorem for self-adjoint operators except at one point. Whenever the square root of a positive unbounded self-adjoint operator is required, the spectral theorem is utilized to establish the existence of the square root. Elementary proofs of the existence of the square root for positive self-adjoint bounded operators are well known (cf., e.g., [2] or [3]). We present here an elementary proof of the existence of the square root for the unbounded case. 2. Let H be a complex Hilbert space with inner product (.,.) and norm 11. A linear operator A with domain DA dense in H possesses an adjoint A* which is a closed linear operator; if A = A*, A is self-adjoint. If (Ax, x) > y2(x, x), Py > 0 for all x E DA, A will be called positive. If (Ax, x) > 0 for all x E DA, X # 0, then A will be called positive definite. If (Ax, x) _ 0 for all x E DA, A will be called positive. We write A ? B if A B is positive. An elementary proof exists for the following theorem: Every positive bounded self-adjoint operator A possesses a unique positive bounded self-adjoint square root which commutes with every bounded operator which commutes with A. We will utilize this result and prove first: THEOREM 1. Every positive self-adjoint operator A possesses a unique positive self-adjoint square root which commutes with every bounded operator that commutes with A. (Here A commutes with a bounded operator T if TA extends AT.) Next we will show: THEOREM 2. Theorem 1 remains true if the word strictly is removed. 3. Proof of Theorem 1. From Schwarz's inequality we see that