Fredholm and Regularized Determinants

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Problem 12.108 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Section 12.3 "Hilbert–Schmidt theorem"

Find the spectrum of the integral operators in L2[0,π]L_{2}[0, \pi ] with kernel

?
(a)

K(t,s)=∑n=1∞2−nsin⁡ntsin⁡nsK(t, s) = \sum_{n = 1}^{\infty } 2^{-n} \sin n t \sin n s;

(b)

K(t,s)=∑n=0∞2−ncos⁡ntcos⁡nsK(t, s) = \sum_{n = 0}^{\infty } 2^{-n} \cos n t \cos n s;

(c)

K(t,s)=∑n=1∞2−ncos⁡ntcos⁡nsK(t, s) = \sum_{n = 1}^{\infty } 2^{-n} \cos n t \cos n s.

Problem 12.109 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Section 12.3 "Hilbert–Schmidt theorem"

Find the spectrum of the integral operators in L2[0,1]L_{2}[0,1] with kernel

?
(a)

K(t,s)=min⁡(t,s)K(t, s) = \min (t, s);

(b)

K(t,s)=max⁡(t,s)K(t, s) = \max (t, s) (compare with problem 11.96). Determine to which Neumann-Schatten classes these operators belong.

References

[БСШ17] Бородин П.А., Савчук А.М., Шейпак И.А. Задачи по функциональному анализу. МЦНМО, 2017.