Commutators and spectral theory

Abstract

Let A and B be bounded operators on a Hilbert space and suppose that

$$C=AB-BA$$

(2.1.1)

Then it is easily shown that ||C|| ≤ 2||A|| ||B|| and, by consideration of simple finite-dimensional examples, that the last inequality can become an equality (with C ≠ 0). Thus, in general, the factor 2 cannot be replaced by a smaller number. In case A and C are self-adjoint and C is semi-definite, the inequality can be refined however. In fact in this case, as will be shown below,

$$\left\| C \right\|\le (2\pi )\left\| A \right\|\left\| B \right\|$$

(2.1.2)

where the constant 2/π it is optimal. In this chapter, some related inequalities will also be derived. In addition, an investigation of the spectrum of A when A is self-adjoint and C ≥ 0 will be made.