When is an selfadjoint operatorvalued matrix with positive semidefinite diagonal elements positive semidefinite as well?

We have $p \in \mathbb{N}$ and $\mathcal{H}$ is a Hilbert space.

• let's consider a matrix $\boldsymbol{\Gamma}_p := (C_{i-j})_{i,j=1, ..., p} \in \mathcal{S_H}^{p\times p}\!\!\,,$ that is a $p\times p$-matrix of Hilbert-Schmidt operators $C_{1-p}, ..., C_{p-1}\colon \mathcal{H} \rightarrow \mathcal{H}$,
• $C_0$ is selfadjoint and positive semidefinite and $\forall l \in \{1, ..., p-1\}\colon$ $C^*_l = C_{-l}$ where $"^*"$ denotes the adjoint of an operator and $||C_0||_{\mathcal{S}_H} \geq ||C_i||_{\mathcal{S_H}}, \forall i \in \{1-p, ..., p-1\}.$

With that given, the matrix

$$ \boldsymbol{\Gamma}_p = \begin{bmatrix} C_{0} & C_{-1} & C_{-2} & \ldots & \ldots &C_{1-p} \\ C_1 & C_0 & C_{-1} & \ddots & & \vdots \\ C_2 & C_1 & \ddots & \ddots & \ddots& \vdots \\ \vdots & \ddots & \ddots & \ddots & C_{-1} & C_{-2}\\ \vdots & & \ddots & C_1 & C_0& C_{-1} \\ C_{p-1}& \ldots & \ldots & C_2 & C_1 &C_0 \end{bmatrix} $$

is selfadjoint, diagonal-dominant (respective $||\cdot||_{\mathcal{S_H}}$) with positive semidefinite diagonal elements.

Questions:

1. Is $\boldsymbol{\Gamma}_p$ positive semidefinit?
2. Could it be even strictly positive definite (but I guess it doesn't hold, since $C_0$ is not necessarily strictly positive definite) ?

Thank you very much!