spectrum of weighted sum of projection operators

Let $P_i$ be a collection of projection operators on some Hilbert space $H$, so that $P_iP_j=0$, and so that $\sum_i P_i=Id$, where $Id$ is the identity operator. Assume that there exists a bounded sequence $\lambda_i$ so that the series $T=\sum_i \lambda_i P_i$ defines a bounded operator on $H$. Is it true that the spectrum of $T$ is precisely $\overline{\{\lambda_i:i=1,2,\dots\}}$?

Of course intuitively one would simply like to invert the operator as follows:For $x=\sum_{i}P_ix$, it would seem that $(T-\lambda)(\sum_i \frac{1}{\lambda_i-\lambda}P_ix)=x$. However, because the projections do not have orthogonal ranges, it is not clear if $(\sum_i \frac{1}{\lambda_i-\lambda}P_ix)$ is actually in $H$.

Any help would be appreciated!