Question

(a) Prove that the operation $M_a[u(x)]=a(x) u(x)$ of multiplication by a continuous function $a(x)$ defines a self-adjoint linear operator on the function space $\mathrm{C}^0[a, b]$ with respect to the $\mathrm{L}^2$ inner product. (b) Is $M_a$ also self-adjoint with respect to the weighted inner product $\langle\langle f, g\rangle\rangle=\int_a^b f(x) g(x) w(x) d x$ ?

    (a) Prove that the operation $M_a[u(x)]=a(x) u(x)$ of multiplication by a continuous function $a(x)$ defines a self-adjoint linear operator on the function space $\mathrm{C}^0[a, b]$ with respect to the $\mathrm{L}^2$ inner product. (b) Is $M_a$ also self-adjoint with respect to the weighted inner product $\langle\langle f, g\rangle\rangle=\int_a^b f(x) g(x) w(x) d x$ ?

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Applied Linear Algebra (Undergraduate Texts in Mathematics)

Applied Linear Algebra (Undergraduate Texts in Mathematics)

Peter J. Olver,… 2nd Edition

Chapter 7, Problem 21

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Step 1

The $\mathrm{L}^2$ inner product for functions $f, g \in \mathrm{C}^0[a, b]$ is defined as: \[ \langle f, g \rangle = \int_a^b f(x) \overline{g(x)} \, dx \] where $\overline{g(x)}$ denotes the complex conjugate of $g(x)$. Show more…

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(a) Prove that the operation $M_a[u(x)]=a(x) u(x)$ of multiplication by a continuous function $a(x)$ defines a self-adjoint linear operator on the function space $\mathrm{C}^0[a, b]$ with respect to the $\mathrm{L}^2$ inner product. (b) Is $M_a$ also self-adjoint with respect to the weighted inner product $\langle\langle f, g\rangle\rangle=\int_a^b f(x) g(x) w(x) d x$ ?

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