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$\begingroup$

let ‎$ 1 < P < + ‎\infty‎ $ and‎

‎‎‎$ T‎ :‎ ‎L‎^{P} [ 0 , ‎1] ‎\longrightarrow ‎L‎^{P} [ 0 , ‎1‎]‎‎‎‎‎‎$‎ ‎$ T ( f ( t)) = ‎\int‎_{0}‎^{1} ( ‎s‎^{2} + ‎t‎^{2} ) ‎f(s)ds‎‎‎‎‎‎‎‎$‎‎‎‎ ‎

How ‎can ‎we ‎show ‎that‎ ‎$ T $ ‎is ‎‎rank ‎2‎?‎‎

How is ‎the Spectrum‎ calculation?

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    Writing down the eigen-value equation you find out that eigenvectors must be a polynomials of degree $2$. – 2017-01-17
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    $\int_0^1(s^2+t^2)f(s)ds=\int_{0}^{1}s^2f(s)ds+t^2\int_{0}^{1}f(s)ds$ has range contained in the linear span of $1$ and $t^2$. – 2017-01-17

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