# Partition of unity. Functional analysis

Need to find partition of unity in case of oparator $A_{f}(x)=(|x-1|+x)f(x)$. Operator $A \in L_{2}[0,2]$ Partition of Unity is set of operators $E_{\lambda}=E((- \infty,\lambda])$, where $E(\Omega)=\chi_{\Omega}(A)$. $\Omega$ is Borel set, and $\chi$ is characteristic function of there( i mean $\Omega$).I know how to find $E_{\lambda}$ if $\lambda \in \sigma_{cont}(A)$, but in this task $\lambda=1 \in \sigma_{point}(A)$. Also I know that if $a<b$, then $E_{a}\le E_{b}$. Please, help me. Thanks in advance

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Maybe you could explain us what you have tried, where you are stuck. – Davide Giraudo May 19 '12 at 14:51
Also, what do you exactly mean by a partition of unity for the operator and $A\in L_2$? – Hui Yu May 19 '12 at 14:54
I edited the question – Dzianis May 20 '12 at 18:58
You mean $(Af)(x) = (|x-1| + x) f(x)$. And $A$ is not a member of $L_2[0,2]$, it is an operator on $L_2[0,2]$. Hint: $E_\lambda$ is the projection on a subspace of $L_2[0,2]$ consisting of functions supported on a certain set related to $\lambda$ and the function $g(x) = |x-1|+x$. – Robert Israel May 20 '12 at 19:17
Thank you/ But I really stupid;) Please, tell me, how to find $E_{1}$ – Dzianis May 20 '12 at 20:14