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The symmetric shifted solution operator is positive and self-adjoint

Statement

Assume Countable Choice. In the symmetric case of The $L^{2}$ operator associated with a symmetric elliptic form, with $Ω$ open and $μ \geq β$ , the shifted solution operator $K_{μ}$ of The shifted elliptic solution operator, regarded on $L^{2} (Ω)$ , is self-adjoint and positive: $(K_{μ} f, g)_{L^{2}} = (f, K_{μ} g)_{L^{2}}, (K_{μ} f, f)_{L^{2}} = a_{μ} (K_{μ} f, K_{μ} f) \geq α ∥ K_{μ} f ∥_{H_{0}^{1}}^{2} \geq 0$ for all $f, g \in L^{2} (Ω)$ , with $(K_{μ} f, f)_{L^{2}} = 0$ if and only if $f = 0$ ; in particular $K_{μ}$ is injective. Moreover $a (K_{μ} f, v) = (f - μ K_{μ} f, v)_{L^{2}}$ for every $v \in H_{0}^{1} (Ω)$ , so $K_{μ} f \in D (L)$ and $L (K_{μ} f) = f - μ K_{μ} f$ . Symmetry of $K_{μ}$ is verified from the form; no self-adjointness of the differential expression is assumed.

Facts & Assumptions

Given: Countable Choice; the symmetric divergence-form case with form $a$ and $μ \geq β$ ; the shifted solution operator $K_{μ}$ and the shifted form $a_{μ} = a + μ (\cdot, \cdot)_{L^{2}}$ ; $f, g \in L^{2} (Ω)$ .

[F1]

Defining identity and coercivity: $a_{μ} (K_{μ} h, v) = (h, v)_{L^{2}}$ for all $v \in H_{0}^{1} (Ω)$ and all $h \in L^{2} (Ω)$ , and $a_{μ} (u, u) \geq α ∥ u ∥_{H_{0}^{1}}^{2}$ with $α = θ / 2$ for $u \in H_{0}^{1} (Ω)$ (The shifted elliptic solution operator, A sufficiently large shift is coercive, Bounded, coercive and symmetric sesquilinear forms).

[F2]

Symmetry: $a (u, v) = \overline{a (v, u)}$ and $(v, u)_{L^{2}} = \overline{(u, v)_{L^{2}}}$ , so $a_{μ}$ is symmetric as well; in particular $a_{μ} (u, u)$ is real (The $L^{2}$ operator associated with a symmetric elliptic form, Bounded, coercive and symmetric sesquilinear forms, The formal adjoint and the adjoint weak Dirichlet problem).

[F3]

Density: $H_{0}^{1} (Ω)$ is dense in $L^{2} (Ω)$ (Smooth compactly supported functions of an open set are dense in $L^{2}$ ).

[F4]

Hilbert adjoints and positivity: an operator $T$ on a Hilbert space is self-adjoint when $(T f, g) = (f, T g)$ for all $f, g$ , and positive when $(T f, f) \geq 0$ (The Hilbert-space adjoint of a bounded operator, Self-adjoint, positive, unitary and normal operators).

[F5]

The operator $L$ and its domain: $u \in D (L)$ with $L u = h$ means $u \in H_{0}^{1} (Ω)$ and $a (u, v) = (h, v)_{L^{2}}$ for all $v \in H_{0}^{1} (Ω)$ (The $L^{2}$ operator associated with a symmetric elliptic form, The associated elliptic operator is densely defined, symmetric and lower bounded).

Proof

technique · direct

1.1F1F2F4givenalgebra

Self-adjointness. By [F1] applied to $g$ and [F2], and then to $f$ , $(K_{μ} f, g)_{L^{2}} = \overline{(g, K_{μ} f)_{L^{2}}} = \overline{a_{μ} (K_{μ} g, K_{μ} f)} = a_{μ} (K_{μ} f, K_{μ} g) = (f, K_{μ} g)_{L^{2}},$ for all $f, g \in L^{2} (Ω)$ ; hence $K_{μ}$ is self-adjoint by [F4].

2.1F1F3F4step 1.1givenalgebra

Positivity and injectivity. Taking $g = f$ in the computation of step 1.1 and using [F1], $(K_{μ} f, f)_{L^{2}} = a_{μ} (K_{μ} f, K_{μ} f) \geq α ∥ K_{μ} f ∥_{H_{0}^{1}}^{2} \geq 0$ . If $(K_{μ} f, f)_{L^{2}} = 0$ , then $α ∥ K_{μ} f ∥_{H_{0}^{1}}^{2} \leq 0$ , so $K_{μ} f = 0$ ; then for every $v \in H_{0}^{1} (Ω)$ the defining identity gives $(f, v)_{L^{2}} = a_{μ} (K_{μ} f, v) = 0$ , and density of $H_{0}^{1} (Ω)$ in $L^{2} (Ω)$ ([F3]) gives $f = 0$ . Conversely $f = 0$ gives $K_{μ} f = 0$ and hence $(K_{μ} f, f)_{L^{2}} = 0$ ; thus $K_{μ}$ is positive and injective.

3.1F1F5givenalgebra∎

Range description. For $f \in L^{2} (Ω)$ and every $v \in H_{0}^{1} (Ω)$ , $a (K_{μ} f, v) = a_{μ} (K_{μ} f, v) - μ (K_{μ} f, v)_{L^{2}} = (f - μ K_{μ} f, v)_{L^{2}},$ because the datum $f - μ K_{μ} f$ lies in $L^{2} (Ω)$ . By [F5] this says $K_{μ} f \in D (L)$ and $L (K_{μ} f) = f - μ K_{μ} f$ .

Depends on

Used by

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Sources

Richard S. Laugesen, Linear Analysis and Partial Differential Equations (University of Illinois, 2020, complete 158-page graduate notes) (standard reference, not scraped)
John K. Hunter, Notes on Partial Differential Equations (UC Davis, revised 18 June 2014, complete 242-page notes) (standard reference, not scraped)
Richard S. Laugesen, Spectral Theory of Partial Differential Equations (University of Illinois lecture notes, arXiv:1203.2344, complete 120 pages) (standard reference, not scraped)