Editing Operator norm (section)

== Operators on a Hilbert space ==

Suppose <math>H</math> is a real or complex [[Hilbert space]]. If <math>A : H \to H</math> is a bounded linear operator, then we have
<math display="block">\|A\|_\text{op} = \left\|A^*\right\|_\text{op}</math>
and
<math display="block">\left\|A^* A\right\|_\text{op} = \|A\|_\text{op}^2,</math>
where <math>A^{*}</math> denotes the [[adjoint operator]] of <math>A</math> (which in [[Euclidean space]]s with the standard [[inner product]] corresponds to the [[conjugate transpose]] of the matrix <math>A</math>).

In general, the [[spectral radius]] of <math>A</math> is bounded above by the operator norm of <math>A</math>:
<math display="block">\rho(A) \leq \|A\|_\text{op}.</math>

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 operator]]s is one class of such examples. A nonzero quasinilpotent operator <math>A</math> has spectrum <math>\{0\}.</math> So <math>\rho(A) = 0</math> while <math>\|A\|_\text{op} > 0.</math>

However, when a matrix <math>N</math> is [[Normal matrix|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
<math display="block">\rho(N) = \|N\|_\text{op}.</math>

This formula can sometimes be used to compute the operator norm of a given bounded operator <math>A</math>: define the [[Hermitian operator]] <math>B = A^{*} A,</math> determine its spectral radius, and take the [[square root of a matrix|square root]] to obtain the operator norm of <math>A.</math>

The space of bounded operators on <math>H,</math> with the [[Topological space|topology]] induced by operator norm, is not [[Separable space|separable]]. 
For example, consider the [[Lp space]] <math>L^2[0, 1],</math> which is a Hilbert space. 
For <math>0 < t \leq 1,</math> let <math>\Omega_t</math> be the [[Indicator function|characteristic function]] of <math>[0, t],</math> and <math>P_t</math> be the [[multiplication operator]] given by <math>\Omega_t,</math> that is,
<math display="block">P_t (f) = f \cdot \Omega_t.</math>

Then each <math>P_t</math> is a bounded operator with operator norm 1 and
<math display="block">\left\|P_t - P_s\right\|_\text{op} = 1 \quad \mbox{ for all } \quad t \neq s.</math>

But <math>\{P_t : 0 < t \leq 1\}</math> is an [[uncountable set]]. 
This implies the space of bounded operators on <math>L^2([0, 1])</math> is not separable, in operator norm. 
One can compare this with the fact that the sequence space <math>\ell^{\infty}</math> is not separable.

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