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Finite orthogonal sums of self-adjoint operators and of bounded operators

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )), the choice hypothesis of the cited definitions of the adjoint. Fix an integer $r \geq 1$ , let $H_{1}, \dots, H_{r}$ be complex Hilbert spaces with inner products linear in the first argument (Hilbert space), and let $H : = ⨁_{l = 1}^{r} H_{l}, ⟨ (x_{l}), (y_{l}) ⟩ : = \sum_{l = 1}^{r} ⟨ x_{l}, y_{l} ⟩_{H_{l}},$ be their orthogonal sum. For each $l$ let $A_{l} : D (A_{l}) \subset H_{l} \to H_{l}$ be a densely defined self-adjoint operator (Adjoint of a densely defined operator, Symmetric, self-adjoint and essentially self-adjoint operators) and put $D (A) : = {(x_{l}) \in H : x_{l} \in D (A_{l}) for all l, \sum_{l = 1}^{r} ∥ A_{l} x_{l} ∥_{H_{l}}^{2} < \infty}, A (x_{l}) : = (A_{l} x_{l}) .$ Then:

(a) $H$ is a complex Hilbert space, $D (A)$ is a dense linear subspace of $H$ , and $A : D (A) \to H$ is a linear operator;

(b) $A$ is symmetric and self-adjoint, and $∥ A x ∥^{2} = \sum_{l = 1}^{r} ∥ A_{l} x_{l} ∥_{H_{l}}^{2}$ for every $x \in D (A)$ ;

(c) if $B_{l} \in B (H_{l})$ for every $l$ (A bounded linear operator between normed spaces), then $B : = ⨁_{l = 1}^{r} B_{l}$ , defined by $B (x_{l}) : = (B_{l} x_{l})$ for all $(x_{l}) \in H$ , is bounded with $∥ B ∥ = \max_{1 \leq l \leq r} ∥ B_{l} ∥$ (The operator norm as the least bound and as the unit-sphere or unit-ball supremum), its adjoint is $B^{*} = ⨁_{l = 1}^{r} B_{l}^{*}$ (The Hilbert-space adjoint of a bounded operator), and $B$ is unitary if every $B_{l}$ is unitary.

No assertion is made about infinite orthogonal sums: only the finite case is proved here.

Facts & Assumptions

Given: The Axiom of Countable Choice; the integer $r \geq 1$ ; complex Hilbert spaces $H_{1}, \dots, H_{r}$ with pairings linear in the first argument; densely defined self-adjoint operators $A_{l}$ on $H_{l}$ ; bounded operators $B_{l} \in B (H_{l})$ ; vectors $x = (x_{l}), y = (y_{l}) \in H$ ; unit vectors $u_{l} \in H_{l}$ ; scalars $α, β \in C$ .

[F1]

The orthogonal sum. $⟨ (x_{l}), (y_{l}) ⟩ = \sum_{l} ⟨ x_{l}, y_{l} ⟩_{l}$ is a Hermitian sesquilinear form, linear in the first argument, with $⟨ x, x ⟩ = \sum_{l} ∥ x_{l} ∥^{2} \geq 0$ and equality exactly for $x = 0$ ; the induced norm is $∥ x ∥^{2} = \sum_{l} ∥ x_{l} ∥^{2}$ (Hilbert space). A complex Hilbert space is complete for its norm.

[F2]

Adjoints. For a densely defined operator $T$ on a Hilbert space, $D (T^{*})$ consists of those $y$ for which $x \mapsto ⟨ T x, y ⟩$ is a bounded linear functional on $D (T)$ , and $T^{*} y$ is its Riesz representative; $T$ is symmetric when $⟨ T x, y ⟩ = ⟨ x, T y ⟩$ for all $x, y \in D (T)$ , and self-adjoint when $D (T^{*}) = D (T)$ and $T^{*} = T$ (Adjoint of a densely defined operator, Symmetric, self-adjoint and essentially self-adjoint operators). In particular a symmetric operator with $D (T^{*}) \subseteq D (T)$ is self-adjoint.

[F3]

Bounded operators. For $B \in B (H)$ one has $∥ B x ∥ \leq ∥ B ∥ ∥ x ∥$ , and $∥ B ∥ = \sup_{∥ x ∥ \leq 1} ∥ B x ∥$ is attained as a supremum over the unit ball (A bounded linear operator between normed spaces, The operator norm as the least bound and as the unit-sphere or unit-ball supremum). The Hilbert adjoint $B^{*}$ is characterized by $⟨ B x, y ⟩ = ⟨ x, B^{*} y ⟩$ for all $x, y \in H$ ; $B$ is unitary exactly when $B^{*} B = B B^{*} = I$ (The Hilbert-space adjoint of a bounded operator).

Proof

technique · direct; verify the Hilbert-space axioms componentwise, prove density, symmetry and the adjoint computation slot by slot, and compute the norm and adjoint of a bounded orthogonal sum

Given: The data of the statement and [F1]-[F3].

1.1F1algebra

[Hilbert-space axioms] [F1] gives sesquilinearity, Hermitian symmetry and positive definiteness of the displayed form, since all of these are checked componentwise and the sum over $l = 1, \dots, r$ is finite. For completeness let $(x^{(n)})_{n}$ be a Cauchy sequence in $H$ , $x^{(n)} = (x_{l}^{(n)})_{l}$ . For each $l$ and all $m, n$ one has $∥ x_{l}^{(n)} - x_{l}^{(m)} ∥_{l} \leq ∥ x^{(n)} - x^{(m)} ∥$ , so $(x_{l}^{(n)})_{n}$ is Cauchy in $H_{l}$ and converges to some $x_{l} \in H_{l}$ ; then $\sum_{l} ∥ x_{l} ∥^{2} = \lim_{n} \sum_{l} ∥ x_{l}^{(n)} ∥^{2} \leq {lim sup}_{n} ∥ x^{(n)} ∥^{2} < \infty$ because a Cauchy sequence is bounded, so $x : = (x_{l}) \in H$ , and $∥ x^{(n)} - x ∥^{2} = \lim_{m} \sum_{l} ∥ x_{l}^{(n)} - x_{l}^{(m)} ∥^{2} = \lim_{m} ∥ x^{(n)} - x^{(m)} ∥^{2}$ , which tends to $0$ as $n \to \infty$ uniformly in $n$ because the sequence is Cauchy; hence $x^{(n)} \to x$ . This proves (a) for the space.

1.2F1F2algebra

[Density and linearity of the domain] The algebraic sum $⨁_{l} D (A_{l}) \subseteq D (A)$ : each $(x_{l})$ with $x_{l} \in D (A_{l})$ has a finite sum $\sum_{l} ∥ A_{l} x_{l} ∥^{2}$ . It is dense in $H$ : given $x \in H$ and $ε > 0$ , density of $D (A_{l})$ gives for each $l$ a vector $y_{l} \in D (A_{l})$ with $∥ x_{l} - y_{l} ∥ < ε / \sqrt{r}$ , so $y : = (y_{l}) \in ⨁_{l} D (A_{l})$ and $∥ x - y ∥^{2} = \sum_{l} ∥ x_{l} - y_{l} ∥^{2} < ε^{2}$ . The set $D (A)$ is a linear subspace because $\sum_{l} ∥ α A_{l} x_{l} + β A_{l} y_{l} ∥^{2} \leq 2 ∣ α ∣^{2} \sum_{l} ∥ A_{l} x_{l} ∥^{2} + 2 ∣ β ∣^{2} \sum_{l} ∥ A_{l} y_{l} ∥^{2} < \infty$ for $x, y \in D (A)$ , and $A$ is linear because each $A_{l}$ is.

1.3F2algebra

[Symmetry] Let $x, y \in D (A)$ . Then $⟨ A x, y ⟩ = \sum_{l} ⟨ A_{l} x_{l}, y_{l} ⟩_{l} = \sum_{l} ⟨ x_{l}, A_{l} y_{l} ⟩_{l} = ⟨ x, A y ⟩$ , because each $A_{l}$ is self-adjoint and hence symmetric; all sums are finite.

1.4F1F2algebra

[The adjoint domain] Let $y \in D (A^{*})$ and $z : = A^{*} y$ . For fixed $l$ and each $u \in D (A_{l})$ let $x^{(l, u)} \in H$ have $u$ in slot $l$ and $0$ in all other slots; then $x^{(l, u)} \in D (A)$ and $A x^{(l, u)}$ has $A_{l} u$ in slot $l$ . Using [F2] for $A$ and the componentwise pairing, $⟨ A_{l} u, y_{l} ⟩_{l} = ⟨ A x^{(l, u)}, y ⟩ = ⟨ x^{(l, u)}, z ⟩ = ⟨ u, z_{l} ⟩_{l}$ for every $u \in D (A_{l})$ . Hence $u \mapsto ⟨ A_{l} u, y_{l} ⟩_{l}$ is bounded on $D (A_{l})$ , represented by $z_{l}$ ; so $y_{l} \in D (A_{l}^{*}) = D (A_{l})$ and $A_{l}^{*} y_{l} = z_{l}$ , that is $A_{l} y_{l} = z_{l}$ . Summing, $\sum_{l} ∥ A_{l} y_{l} ∥^{2} = \sum_{l} ∥ z_{l} ∥^{2} = ∥ z ∥^{2} < \infty$ , so $y \in D (A)$ and $A y = z$ . Hence $D (A^{*}) \subseteq D (A)$ and $A^{*} y = A y$ .

2.1step 1.1step 1.2step 1.3step 1.4F1F2algebra

[Self-adjointness] Step 1.3 gives symmetry; symmetry gives $D (A) \subseteq D (A^{*})$ with $A^{*}$ agreeing with $A$ there (Adjoint of a densely defined operator), and step 1.4 gives the converse inclusion, so $A^{*} = A$ by [F2]. The norm identity is [F1] applied to $A x = (A_{l} x_{l})$ . This proves (b).

3.1F3step 1.1algebra∎

[Bounded sums] Let $B_{l} \in B (H_{l})$ and put $M : = \max_{1 \leq l \leq r} ∥ B_{l} ∥$ , which exists because the index set is finite and nonempty. For $x \in H$ , $∥ B x ∥^{2} = \sum_{l} ∥ B_{l} x_{l} ∥^{2} \leq M^{2} \sum_{l} ∥ x_{l} ∥^{2} = M^{2} ∥ x ∥^{2}$ , so $B$ is bounded with $∥ B ∥ \leq M$ by [F3]. If $M > 0$ , choose $l_{0}$ with $∥ B_{l_{0}} ∥ = M$ ; since $∥ B_{l_{0}} ∥ > 0$ , $H_{l_{0}} \neq {0}$ . For each $η > 0$ , the unit-sphere formula for the operator norm in [F3] gives a unit vector $u_{l_{0}} \in H_{l_{0}}$ with $∥ B_{l_{0}} u_{l_{0}} ∥ > M - η$ . The vector with $u_{l_{0}}$ in slot $l_{0}$ and $0$ elsewhere is a unit vector of $H$ , so $∥ B ∥ \geq ∥ B x ∥ = ∥ B_{l_{0}} u_{l_{0}} ∥ > M - η$ ; letting $η ↓ 0$ gives $∥ B ∥ \geq M$ . If $M = 0$ , the upper bound already gives $∥ B ∥ = 0 = M$ . Thus $∥ B ∥ = \max_{l} ∥ B_{l} ∥$ . For $B_{l}^{*}$ the adjoint of $B_{l}$ , the operator $⨁_{l} B_{l}^{*}$ satisfies $⟨ B x, y ⟩ = \sum_{l} ⟨ B_{l} x_{l}, y_{l} ⟩_{l} = \sum_{l} ⟨ x_{l}, B_{l}^{*} y_{l} ⟩_{l} = ⟨ x, (⨁_{l} B_{l}^{*}) y ⟩$ for all $x, y \in H$ , so $B^{*} = ⨁_{l} B_{l}^{*}$ by [F3]. If every $B_{l}$ is unitary then $B^{*} B = ⨁_{l} B_{l}^{*} B_{l} = ⨁_{l} I_{H_{l}} = I$ and likewise $B B^{*} = I$ , so $B$ is unitary. This proves (c).

Remarks

Where the finiteness is used. Both the norm identity of (b) and the norm formula of (c) are finite-sum statements; for infinite orthogonal sums the domain condition in (b) and the supremum in (c) require separate arguments, and no such statement is made here.
Choice. Only the Axiom of Countable Choice is declared, as the choice hypothesis of the cited definitions of the adjoint; the argument itself handles finitely many slots and makes no further selection.

Depends on

Dependency tree · 2 levels

Sources

G. Teschl, Mathematical Methods in Quantum Mechanics, 1st edition 2009, Section 2.5, printed pages 79-80 (PDF pages 91-92): orthogonal sums of operators, reducing subspaces and the adjoint of an orthogonal sum; Section 1.4, printed page 45 (PDF page 57): orthogonal sums of Hilbert spaces