Reid's inequality

The reidsche inequality is a mathematical inequality in the field of operator theory in Hilbert spaces . It was proven in 1951 by William Thomas Reid .

formulation

Be a Hilbert space and continuous linear operators on , so that: ${\displaystyle H}$ ${\displaystyle A,B}$ ${\displaystyle H}$

• ${\displaystyle A}$is a positive operator , that is, for all ${\displaystyle \langle Ax,x\rangle \geq 0}$ ${\displaystyle x\in H}$
• ${\displaystyle AB}$is self-adjoint , that is, for everyone . ${\displaystyle \langle ABx,y\rangle =\langle x,ABy\rangle }$ ${\displaystyle x,y\in H}$

Then applies to everyone . ${\displaystyle |\langle ABx,x\rangle |\leq \|B\|\langle Ax,x\rangle }$ ${\displaystyle x\in H}$

The proof can be done by elementary means, that is, without spectral theory or functional calculus . Essentially, it is a clever application of the Cauchy-Schwarz inequality for positive semidefinite sesquilinear on . ${\displaystyle (x,y)\to \langle Ax,y\rangle }$ ${\displaystyle H}$

application

• If the positive operators and are interchangeable, that is , then is also positive. ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle AB=BA}$ ${\displaystyle AB}$

As a proof, let the identical operator be on the underlying Hilbert space. Without limitation is . Then is also and therefore . Since also interchanged with , it is self adjoint, and Reid's inequality yields . So is , that is . ${\displaystyle I}$ ${\displaystyle \|B\|\leq 1}$ ${\displaystyle 0\leq I-B\leq 1}$ ${\displaystyle \|I-B\|\leq 1}$ ${\displaystyle A}$ ${\displaystyle I-B}$ ${\displaystyle A(I-B)}$ ${\displaystyle \langle A(I-B)x,x\rangle \leq \|I-B\|\langle Ax,x\rangle \leq \langle Ax,x\rangle }$ ${\displaystyle A-AB=A(I-B)\leq A}$ ${\displaystyle AB\geq 0}$

The demonstration of this important result with the help of Reid's inequality requires only elementary aids. With advanced theory one can get this result just as quickly. Then one considers the C * -algebra generated by and , which, since commutative, is isomorphic to an algebra of continuous functions on a locally compact Hausdorff space according to the Gelfand-Neumark theorem , and the above application is reduced to the fact that the product of two continuous functions Functions is such a function again. ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle X}$ ${\displaystyle X\mapsto \mathbb {R} _{0}^{+}}$

swell

• WT Reid: Symmetrizable completely continuous linear transformations in Hilbert space , Duke Mathematical Journal, Volume 18, Pages 41-56, (1951)
• Harro Heuser: functional analysis , Teubner-Verlag, 1975