Let me introduce the term {$E_\lambda:\lambda \geq0$} is the spectral resolution of identity of a self adjoint densely defined, positive and closed operator $A:D(A)\subset X\rightarrow X$ , Where X be the Hilbert space. And $f:[0,T]\rightarrow X$ is a function having eigenfunction expression $f(t)=\int_{0}^{T}exp(2\lambda s)d||E_\lambda f(s)||^{2}ds <\infty$ and $A_\epsilon$ is a positive real number. The expression above the blue box is fine. But I can not understand, how the expression in blue box is obtained and what is the meaning of the expression in sky colour box in Hilbert space. I have god this expression in time of reading a paper on back ward heat conduction problem namely ''A simple regularization method for ill-posed evolution equation '' by Nguyen Huy Tuan, Dang Duc Trong, Ho Chi Minh City. Please help me to understand the things.
Starting with the first line, \begin{align} u(t) - u_\epsilon(t) & = \int_{A_\epsilon}^\infty e^{-\lambda t} d E_\lambda \left( e^{\lambda T} \varphi - \int_t^T e^{\lambda s} f(s) ds \right) \end{align} so the squared norm of this expression is (recall that $E_\lambda$ is a projector on the Hilbert space) \begin{align} \| u(t) - u_\epsilon(t) \|^2 & = \int_{A_\epsilon}^\infty e^{-2\lambda t} \left\| d E_\lambda \left( e^{\lambda T} \varphi - \int_t^T e^{\lambda s} f(s) ds \right) \right\|^2 . \end{align} The notation would make more sense if the spectral measure was discrete, which I recommend writing down if you are confused about it.
So it seems to me that the estimate in the first box just contains a bunch of extra terms, and $(\cdots)^2$ means $\|\cdots\|^2$.
The second box is obtained from this expression by setting $\varepsilon = 0$ and $t = 0$ (I am guessing from the first expression that $u_\varepsilon = 0$ for $\varepsilon = 0$).