References  SPDEs2016/17ExerciseSheet4

University of Freiburg

SPDEs 2016/17 Exercise Sheet 4

Lecture and exercises: Philipp Harms, Tolulope Fadina Due date: November 16, 2016

4.1. Regularity of diagonal operators on Hilbert spaces

LetH be a Hilbert space with orthonormal basisB, letl :B!Rbe a function, and let T :D(T)✓H!H be the diagonal linear operator given by

T b=lbb, D(T) = (

h2H:

Â

b2B|lb|²hb,hi²H<• )

.

We consider Bas a measure space with the counting measure #. Show the following statements hold:

a) T 2L(H)iffl 2L^•(B), andkTkL(H)=klkL^•(B). b) T 2L₁(H)iffl 2L¹(B), andkTkL₁(H)=klkL¹(B). c) T 2L₂(H)iffl 2L²(B), andkTkL₂(H)=klk_L²_(B).

Hint: While this is not necessary, you might find it convenient to represent elements of the completed tensor product not by equivalence classes of Cauchy sequences, but as in [Rya02, Proposition 6.10]: anyu2E⌦ˆgpF can be represented as a convergent series u=Â^•_n=1x_n⌦y_nsuch thatk(x_n)k^w_p0k(y_n)kp is finite and arbitrarily close tokukE⌦ˆgpF.

References

[Rya02] Raymond A. Ryan.Introduction to Tensor Products of Banach Spaces. Springer Monographs in Mathematics. Springer, 2002.