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The only reasonable condition I can think of is that $P = g(A)$ where $g \ge 1$ is a bounded continuous function on $\mathbb R$. It is instructive to consider the case where $A$ is the multiplicatio …

The spectral theorem is for normal operators (in particular, hermitian, or real symmetric, matrices). If $A$ is a real matrix, $A A^T$ is a real symmetric matrix.

For any compact set K of complex numbers disjoint from the spectrum of T, there is $\epsilon > 0$ such that for every operator S with $\|S-T\| < \epsilon$, K is disjoint from the spectrum of $S$. Nam …

By "$I$ is well-defined", I presume you mean you have a continuous injection $\iota$ of $X$ into $Y$. If $X$ and $Y$ are not isomorphic it will not be a bijection, because there is no continuous line …

The spectrum of an operator is always a closed set. But perhaps they are defining the "continuous spectrum" to be all points of the spectrum that are not in the point spectrum (i.e. not eigenvalues). …

Let $f$ be your function that is $0$ in a disk $D_0$ around $0$ and $1$ in a disk $D_1$ around $1$. There is (by Runge) a sequence of polynomials $g_n$ such that $g_n \to f$ uniformly on $D_0 \cup D_ …

The most important thing about residual spectrum, I think, is that some classes of operators, e.g. normal operators on Hilbert space, don't have any.

Let's assume the matrix $A$ has all its row sums equal to $\lambda$, the largest eigenvalue. We can rescale the rows and columns of any other nonnegative irreducible matrix by a similarity transformat …

My guess is that this is referring to the resolution of the identity corresponding to the operator, which is sometimes called the "spectral decomposition" (e.g. in Rudin, Functional Analysis) or "spec …

The set of $n \times n$ matrices with positive permanent is connected. Given such a matrix $A$, consider the expansion of $P(A)$ in minors along the first row: say $P(A) = \sum_{j=1}^n a_{1j} P(A_{1 …

No: given $\eta > \epsilon > 0$, there are $n \times n$ symmetric matrices $A_n$ with spectral radius $> \eta$, such that $Pr\left[|v^TA_nv|/\|v\|^2 > \epsilon\right] < e^{-cn}$ for some $c > 0$. I a …

Upper semicontinuity of the spectrum is the following statement: if $U$ is a neighbourhood of $\text{Sp}(x)$, then there is a neighbourhood $V$ of $x$ such that $\text{Sp}(y) \subset U$ for all $y \in …

For example, consider the Laplacian on $[-1,1] \times [-1,1]$ with Dirichlet boundary conditions. If $m$ and $n$ are distinct positive integers, $\sin(m \pi x) \sin(n \pi y)$ and $\sin(n \pi x) \sin( …

For example, if $$ \eqalign{A &= \pmatrix{1 & t\cr 1 & 1\cr},\ v = \pmatrix{\sqrt{t}\cr 1},\ u =\pmatrix{1\cr \sqrt{t}},\cr \frac{(u^T v)^2}{(u^T u)(v^T v)} &= \frac{4t}{(1+t)^2} \to 0 \ \text{as}\ t …

No. For example, the spectral radii of $$ A = \pmatrix{0 & 1 & 1\cr 1 & 0 & 1\cr 0 & 0 & 0\cr},\ A' = \pmatrix{0 & 0 & 3\cr 1 & 0 & 1\cr 0 & 0 & 0\cr}$$ are $1$ and $0$ respectively.