Spectral Theory in Hilbert Space
Spectral Theory in Hilbert Space
Spectral Theory in Hilbert Space
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26 4. OPERATORS<br />
clearly uniquely determ<strong>in</strong>ed by v ∈ H2, if it exists. It is also obvious<br />
that v ∗ depends l<strong>in</strong>early on v, so we def<strong>in</strong>e D(T ∗ ) to be those v ∈ H2<br />
for which we can f<strong>in</strong>d a v ∗ ∈ H1, and set T ∗ v = v ∗ . There is no reason<br />
to expect the adjo<strong>in</strong>t T ∗ to be densely def<strong>in</strong>ed. In fact, we may have<br />
D(T ∗ ) = {0}, so T ∗ may not itself have an adjo<strong>in</strong>t. To understand this<br />
rather confus<strong>in</strong>g situation it turns out to be useful to consider graphs<br />
of operators.<br />
The graph of T is the set GT = {(u, T u) | u ∈ D(T )}. This set is<br />
clearly l<strong>in</strong>ear and may be considered a l<strong>in</strong>ear subset of the orthogonal<br />
direct sum H1 ⊕ H2, consist<strong>in</strong>g of all pairs (u1, u2) with u1 ∈ H1 and<br />
u2 ∈ H2 with the natural l<strong>in</strong>ear operations and provided with the scalar<br />
product 〈(u1, u2), (v1, v2)〉 = 〈u1, v1〉1 + 〈u2, v2〉2. This makes H1 ⊕ H2<br />
<strong>in</strong>to a <strong>Hilbert</strong> space (Exercise 4.6).<br />
We now def<strong>in</strong>e the boundary operator U : H1 ⊕ H2 → H2 ⊕ H1 by<br />
U(u1, u2) = (−iu2, iu1) (the term<strong>in</strong>ology is expla<strong>in</strong>ed <strong>in</strong> Chapter 9). It<br />
is clear that U is isometric and surjective (onto H2 ⊕ H1). It follows<br />
that U is unitary. If H1 = H2 = H it is clear that U is selfadjo<strong>in</strong>t and<br />
<strong>in</strong>volutary (i.e., U 2 is the identity). Now put<br />
(4.1) (GT ) ∗ := U((H1 ⊕ H2) ⊖ GT ) = (H2 ⊕ H1) ⊖ UGT .<br />
The second equality is left to the reader to verify who should also<br />
verify that (GT ) ∗ is a graph of an operator (i.e., the second component<br />
of each element <strong>in</strong> (GT ) ∗ is uniquely determ<strong>in</strong>ed by the first) if and<br />
only if T is densely def<strong>in</strong>ed. If T is densely def<strong>in</strong>ed we now def<strong>in</strong>e T ∗<br />
to be the operator whose graph is (GT ) ∗ . This means that T ∗ is the<br />
operator whose graph consists of all pairs (v, v ∗ ) ∈ H2 ⊕ H1 such that<br />
〈T u, v〉2 = 〈u, v ∗ 〉1 for all u ∈ D(T ), i.e., our orig<strong>in</strong>al def<strong>in</strong>ition. An<br />
immediate consequence of (4.1) is that T ⊂ S implies S ∗ ⊂ T ∗ .<br />
We say that an operator is closed if its graph is closed as a subspace<br />
of H1 ⊕ H2. This is an important property; <strong>in</strong> many ways the<br />
property of be<strong>in</strong>g closed is almost as good as be<strong>in</strong>g bounded. An everywhere<br />
def<strong>in</strong>ed operator is actually closed if and only if it is bounded<br />
(Exercise 4.7). It is clear that all adjo<strong>in</strong>ts, hav<strong>in</strong>g graphs that are orthogonal<br />
complements, are closed. Not all operators are closeable, i.e.,<br />
have closed extensions; for this is required that the closure GT of GT<br />
is a graph. But it is clear from (4.1) that the closure of the graph is<br />
(GT ∗)∗ . So, we have proved the follow<strong>in</strong>g proposition.<br />
Proposition 4.3. Suppose T is a densely def<strong>in</strong>ed operator <strong>in</strong> a<br />
<strong>Hilbert</strong> space H. Then T is closeable if and only if the adjo<strong>in</strong>t T ∗ is<br />
densely def<strong>in</strong>ed. The smallest closed extension (the closure) T of T is<br />
then T ∗∗ .<br />
The proof is left to Exercise 4.9. Note that if T is closed, its doma<strong>in</strong><br />
D(T ) becomes a <strong>Hilbert</strong> space if provided by the scalar product<br />
〈u, v〉T = 〈u, v〉1 + 〈T u, T v〉2.