Operator Norm - Operators On A Hilbert Space

Operators On A Hilbert Space

Suppose H is a real or complex Hilbert space. If A : H → H is a bounded linear operator, then we have

and

where A* denotes the adjoint operator of A (which in Euclidean Hilbert spaces with the standard inner product corresponds to the conjugate transpose of the matrix A).

In general, the spectral radius of A is bounded above by the operator norm of A:

To see why equality may not always hold, consider the Jordan canonical form of a matrix in the finite dimensional case. Because there are non-zero entries on the superdiagonal, equality may be violated. The quasinilpotent operators is one class of such examples. A nonzero quasinilpotent operator A has spectrum {0}. So ρ(A) = 0 while ||A||_op > 0.

However, when a matrix N is normal, its Jordan canonical form is diagonal (up to unitary equivalence); this is the spectral theorem. In that case it is easy to see that

The spectral theorem can be extended to normal operators in general. Therefore the above equality holds for any bounded normal operator N. This formula can sometimes be used to compute the operator norm of a given bounded operator A: define the Hermitian operator B = A*A, determine its spectral radius, and take the square root to obtain the operator norm of A.

The space of bounded operators on H, with the topology induced by operator norm, is not separable. For example, consider the Hilbert space L2. For 0 < t ≤ 1, let Ω_t be the characteristic function of, and P_t be the multiplication operator given by Ω_t, i.e.

Then each P_t is a bounded operator with operator norm 1 and

But {P_t} is an uncountable set. This implies the space of bounded operators on L2 is not separable, in operator norm. One can compare this with the fact that the sequence space l ∞ is not separable.

The set of all bounded operators on a Hilbert space, together with the operator norm and the adjoint operation, yields a C*-algebra.