An upper bound for the largest eigenvalue of a positive semidefinite block banded matrix

The new upper bound
$$ \lambda_\mathrm{max}(A)\le\sum_{k=1}^{p+1}\max_{i\equiv k\pmod{p+1}}\lambda_\mathrm{max}(A_{ii}) $$
for the largest eigenvalue of a Hermitian positive semidefinite block banded matrix $A=(A_{ij})$ of block semibandwidth $p$ is suggested. In the special case where the diagonal blocks of $A$ are identity matrices, the latter bound reduces to the bound $\lambda_\mathrm{max}(A)\le p+1$, depending on $p$ only, which improves the bounds established for such matrices earlier and extends the bound $\lambda_\mathrm{max}(A)\le2$, old known for $p=1$, i.e., for block tridiagonal matrices, to the general case $p\ge1$.