I will be very grateful if someone tell me why $\|(1-e^{t L})(t L)^{-b/2}(t L)^{b/2}\|_2$ is controled by $\|(t L)^{b/2}\|_2$ were $L$ is a self-adjoint operator and $t>0$.
Please see here https://reader.elsevier.com/reader/sd/pii/S0022247X06010377?token=14DF70907A3A1D7C9769738A2DC68D24527707D5082CED7B5341CCFED2952492448FCC435FD72F1A377E3A499AC124EB&originRegion=eu-west-1&originCreation=20210809210201 (Proof of theorem 3.1.) My Proof. One has $L=\int \lambda d\lambda$ so $\|(1-e^{t L})(t L)^{-b/2}(t L)^{b/2}\|^2_2=\int |(1-e^{t \lambda})(t \lambda)^{-b/2}(t \lambda)^{b/2}|^2 d\lambda\leq C\int (t \lambda)^{b/2}|^2 d\lambda=C\|(t L)^{b/2}\|^2_2$. Is it true?