Doesn't $A=\sum_{j\in\mathbb N}\lambda_j\phi_j\otimes\phi_j$ imply $A$ is bounded?

Question

In the book An Introduction to Computational Stochastic PDEs, you can find the following assumption:

Doesn't this already imply that $A$ is bounded and hence $\mathcal D(A)=H$? By assumptio, we can write $$A=\sum_{j\in\mathbb N}\lambda_j\phi_j\otimes\phi_j\tag1$$ and the right-hand side should be bounded; at least if $\sum_{j\in\mathbb N}\lambda_j<\infty$; or is this the crucial point?

Answer 1

If $$ A=\sum_{j\in\mathbb N}\lambda_j\phi_j\otimes\phi_j $$ with the $\{\phi_j\}$ orthonormal, then $A$ is bounded if and only if the sequence $\{\lambda_j\}$ is bounded. In fact, $$\tag1 \|A\|=\sup\{\lambda_j:\ j\}. $$ This is easily seen. Indeed, any $\xi\in H$ can be written as $\xi=\sum_jx_j\,\phi_j$. Then, if $c$ is the supremum in $(1)$,
$$ \|A\xi\|^2=\Big\|\sum_j x_j\lambda_j\,\phi_j\Big\|^2=\sum_j|x_j|^2\,\lambda_j^2\leq c^2\,\sum_j\|x_j|^2=c^2\,\|x\|^2. $$ So $\|A\|\leq c$. Conversely, if $\{\lambda_{j_k}\}$ is a subsequence such that $\lambda_{j_k}\to c$, then $$ \|A\phi_{j_k}\|=\lambda_{j_k}\to c, $$ which shows that $\|A\|\geq c$.

The condition $\sum_j\lambda_j<\infty$ is what one calls for $A$ to be trace-class. It does imply that $A$ is bounded, but there are many bounded operators which are not trace-class.

It is also important to know that a bounded operator that has an orthonormal basis of eigenvectors is normal, and selfadjoint if the eigenvalues are real.

Answer 2

Consider the differential operator $A = \frac{d^2}{dx^2}$ acting on the Hilbert space $L^2([0, \pi])$. The domain $D(A)$ of $A$ is defined as the set of all functions $f \in L^2([0, \pi])$ that are twice differentiable and satisfy the Dirichlet boundary conditions $f(0) = f(\pi) = 0$. The operator $A$ is not defined on the entirety of $L^2([0, \pi])$ because not all functions in this space satisfy these conditions.

The eigenfunctions of this operator are the sine functions $\sin(n x)$ for $n = 1, 2, 3, \ldots$, and the corresponding eigenvalues are $-n^2$. These sine functions form an orthonormal basis for $L^2([0, \pi])$ with respect to the $L^2$ inner product. Any function $f$ in the domain $D(A)$ can be expressed as an infinite series of these eigenfunctions:

$$ f(x) = \sum_{n=1}^{\infty} c_n \sin(nx) $$

where $c_n$ are the coefficients obtained from the Fourier sine series of $f$.