Measure, Integration & Real Analysis · Problem 10B.9

Linear Maps on Hilbert Spaces — Sheldon Axler

Mathematics — Measure Theory · complexification-self-adjoint

Problem

Suppose

V

is a real Hilbert space and

T \in \mathcal{B}(V)

. The complexification of

T

is the function

T_{\mathbf{C}} : V_{\mathbf{C}} \to V_{\mathbf{C}}

defined by

T_{\mathbf{C}}(f + ig) = Tf + iTg

for

f, g \in V

(see Exercise 4 in Section 8B for the definition of

V_{\mathbf{C}}

). ((a)) Show that

T_{\mathbf{C}}

is a bounded operator on the complex Hilbert space

V_{\mathbf{C}}

and

\|T_{\mathbf{C}}\| = \|T\|

. ((b)) Show that

T_{\mathbf{C}}

is invertible if and only if

T

is invertible. ((c)) Show that

(T_{\mathbf{C}})^* = (T^*)_{\mathbf{C}}

. ((d)) Show that

T

is self-adjoint if and only if

T_{\mathbf{C}}

is self-adjoint. ((e)) Use the previous parts of this exercise and 10.49 and 10.38 to show that if

T

is self-adjoint and

V \neq \{0\}

, then

\mathrm{sp}(T) \neq \emptyset