The following question arose while studying the proof of proposition 2.36 in "Heat Kernels and Dirac Operators":
Let $H$ be a Hilbert space and $A$ a self-adjoint operator defined on a subspace of $H$. If there exists an eigenbasis - i.e. a Hilbert $(v_n)_{n\in\mathbb N}$ such that for each $n$ $Av_n=\lambda_nv_n$ for some scalar $\lambda_n$ - can we say that the operator is bounded or even compact? I thought that the eigenbasis may allow us to write $A$ as an infinite sum of projectors and that this may imply the desired result, but I am an amateur when it comes to functional analysis...