Exercise 11.3: Representation of Hilbert-Schmidt operators (2+3+1=6 points) Recall Exercise 9.3. Consider a measure spac

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Exercise 11 3 Representation Of Hilbert Schmidt Operators 2 3 1 6 Points Recall Exercise 9 3 Consider A Measure Spac 1 (58.25 KiB) Viewed 51 times

Exercise 11.3: Representation of Hilbert-Schmidt operators (2+3+1=6 points) Recall Exercise 9.3. Consider a measure space (2,2,) and an orthonormal basis (ej)jen of L² (μ) : L²(0; μ). For any given integral kernel x L²(μμ) : L²(SxS; μμ) one defines a linear mapping T: L²(μ)→ L²(μ) by Tf:= n(w) - f(w) μ(dw) - [K(₁,4). for all fEL²(μ). Note that the operator T is well-defined and continuous - one sees this by means of the Hölder inequality and Fubini's theorem using square-integrability of the kernel K. (a) Show that the mapping : (L² (μ®μ), |-|L² (p®µ)) →→→ (L₂ (L² (μ)), Il·lus), T -> is a well-defined isometry, i. e. prove ΣENIT() = x²(μμ) <0. (b) Prove that for every operator TEL(L² (μ)) of finite rank there exists an integral kernel KEL² (μμ) such that T = T (c) Conclude that every Hilbert-Schmidt operator T on L²(μ) is given by an integral operator T for some K E L² (μμ). Hint: Regarding part (a), recall Parseval's inequality, see e. g. [Gro22, Satz 5.8 (iv)]. For part (c), consider using part (e) of Exercise 9.3.