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Spectrum, resolvent, numerical range, functional calculus, operator semigroups. Special classes of operators: compact, Fredholm, dissipative, differential, integral, pseudodifferential, etc.
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Diagonaloperators on $l^2(N)$ [closed]
Let $D$ be an operator defined by $D(c_n)_{n\in\mathbb{N}} = (a_n c_n)_{n\in\mathbb{N}}$. It's a well known fact that $D$ is well defined as an operator from $l^2(\mathbb{N})$ to $l^2(\mathbb{N})$ if …