Integration with respect to the Borel measure on the unit sphere

Question

Let $\mathcal{L}(E)$ the algebra of all bounded linear operators on a complex Hilbert space $E$.

Let $B_n$ be the open unit ball of $\mathbb{C}^n$ and $\sigma$ be the normalized positive Borel measure on the unit sphere $\partial B_n$.

By this (book), we have $$\int_{\partial B_n} |\lambda_i|^2 d \sigma(\lambda)=\frac1n,$$ and $$\int_{\partial B_n} \lambda_i\overline{\lambda_j} d \sigma(\lambda)=0,\;\;\forall i,j\in\{1,...,n\}\;\text{with}\;i\neq j. $$

I don't understand this notation $\int_{\partial B_n} |\lambda_i|^2 d \sigma(\lambda)$ i.e. here is $\lambda=(\lambda_1,\cdots,\lambda_n)\in \partial B_n$?

Also

If $T_1,\cdots,T_n\in \mathcal{L}(E)$, why $$\sup_{(\lambda_1,\cdots,\lambda_n)\in \partial B_n}\langle (\sum_{i=1}^n \lambda_i T_i)h, (\sum_{i=1}^n \overline{\lambda_i} T_i^*)h \rangle\geq \int_{\partial B_n}\sum_{i,j=1}^n\lambda_i\overline{\lambda_j}\langle T_iT_j^* h, h\rangle d \sigma(\lambda)?$$

Answer 1

Your interpretation of the notation in the integral is correct, $\lambda = (\lambda_1, \dots, \lambda_n)$. This is a completely standard notation for elements of $\mathbb{R}^d$.

For the other part of the question, notice that \begin{align} \sup_{(\lambda_1,\cdots,\lambda_n)\in \partial B_n}\langle (\sum_{i=1}^n \lambda_i T_i)h, (\sum_{i=1}^n \overline{\lambda_i} T_i^*)h \rangle &= \sup_{(\lambda_1,\cdots,\lambda_n)\in \partial B_n} \sum_{i,j = 1}^n \lambda_i \overline{\lambda_j} \langle T_i h, T_j^*h \rangle \\ \end{align} Now notice that \begin{align} \int_{\partial \mathbb{B}^n} \sum_{i,j=1}^n \mu_i \overline{\mu_j} \langle T_i h, T_j^* h \rangle d \sigma(\mu) &\leq \int_{\partial \mathbb{B}^n} \sup_{(\lambda_1,\cdots,\lambda_n)\in \partial B_n} \sum_{i,j=1}^n \lambda_i \overline{\lambda_j} \langle T_i h, T_j^* h \rangle d \sigma(\mu) \\& = \sup_{(\lambda_1,\cdots,\lambda_n)\in \partial B_n} \sum_{i,j=1}^n \lambda_i \overline{\lambda_j} \langle T_i h, T_j^* h \rangle \end{align} since $\sigma$ is normalised such that $\sigma(\partial \mathbb{B}^n) =1$. This is the desired inequality.