Denote by $H=L^2([0,1])$ the Hilbert space of square integrable real valued functions on the interval $[0,1]$. Does there exist a nontrivial smooth real valued function $k$ on the unit interval such that the bounded linear operator $T:H \to H$ defined by $$ Tf(x)= \int_0^x k(x-s) f(s) ds$$ is a normal operator?
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1$\begingroup$ And you probably wanted to exclude $k=0$. $\endgroup$– Christian RemlingCommented Sep 7, 2022 at 17:31
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3$\begingroup$ Such an operator would be pointwise dominated by a multiple of the Volterra operator. Doesn't this imply that the spectral radius is zero? $\endgroup$– Giorgio MetafuneCommented Sep 7, 2022 at 17:45
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