Notes for "Deep null space learning for inverse problems: convergence analysis and rates"

J. Schwab, S. Antholzer, M. Haltmeier, Deep null space learning for inverse problems: convergence analysis and rates, Inverse Problems, 35, 2019, 025008.

Page 8, proof of Theorem 2.8

We have and

Notes:
In the following, we assume is a compact, bounded linear operator. Hence, we immediately have

Since is a piecewise continuous function, according to formula (2.43) in

H. W. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems, vol 375, 1996

we obtain

Hence, we find that which implies . Similarly, we need to show . Assume has an eigen-system . For each , there exists a such that . Let , we obtain

which implies for every . Now, we can give the following calculations

The above equality implies . Obviously, we already proved .

Page 7, proof of Proposition 2.7

We have and,

Notes:
In general, I can not understand that why this equality holds true. If with , I can deduce this equality by using similar deductions in note "Page 8, proof of Theorem 2.8 ".