Articles of infinite matrices

Do eigenvalues of a linear transformation over an infinite dimensional vector space appear in conjugate pairs?

While attempting to answer a question here (namely, the finite dimensional case of the title question: Prove that if $\lambda$ is an eigenvalue of $T$, a linear transformation whose matrix representation has all real entries, then $\overline{\lambda}$ is an eigenvalue of $T$), I noticed the asker did not specify a finite dimensional vector space. Though […]

What is the family of generating functions for the *rows* of this Stirling-number matrix for whose columns they are $\exp(\exp(x)-1)-1 $?

Remark: I give much background because it might significantly help to find an idea how to generate a solution I’m analyzing properties of a certain infinite matrix $U$, for whose columns we have the generating functions $$f_c(x) = (\exp(x)-1)^c \qquad \qquad c=-\infty…+\infty $$, assuming $c$ from $-\infty$ to $+\infty$ (This is the type of Carleman-matrices […]

Frechet derivative of shift operator in $l_2$?

Let $x \in l_2$ and $J(x) = \sum_{k = 1}^{+\infty} x_k x_{k + 1}$. Find $DJ(u)$ and $D(DJ(u))$. Attempted solution Since $x \in l_2$, then $\sum_{k = 1}^{+\infty}x_k < \infty$. Another fact: $\|x\|_{l_2} = (\sum_{k=1}^{+\infty}x_k^{2})^{\frac{1}{2}}$. By the definition of Frechet derivative: $$ \lim_{h\to0} \frac{|J(u + h) – J(u) – (DJ(u))(h)|}{\|h\|_{l_2}} = \lim_{h\to0} \frac{|\sum_{k = 1}^{+\infty}(u_k […]

With infinite size, we can have $P \cdot M = M \cdot D $ (D diagonal) but where $M^{-1}$ does not exist. Can we say “P is diagonalizable”?

(I had this question in mind for longer time, but it is just triggered now by some comments at that recent question in mse) (Background) I was looking at properties of the Pascal-matrix: $\qquad \qquad P_{n=5}=\small \begin{bmatrix} 1 & 1 & 1 & 1 & 1 \\ . & 1 & 2 & 3 & […]