Suppose I have positive semidefinite matrices $A$ and $B$. Then

$$\begin{bmatrix} A & X\\ X & B\end{bmatrix} \succeq 0$$

for $X = A^{\frac 12} C B^{\frac 12}$, where $C$ is the contraction matrix with maximum eigenvalue less than $1$.

Horn, Roger A.; Johnson, Charles R., Topics in matrix analysis, Cambridge etc.: Cambridge University Press. viii, 607 p. \sterling 45.00; \$ 59.50 (1991). ZBL0729.15001.

I have some questions:

1. Is it possible for $C$ to have negative eigenvalues?

2. Are their any properties of $C$ other than eigenvalue $< 1$? Please, suggest a book or something.

3. Is it possible to compute the maximum bounds of the matrix $C$ (Where any random contraction is inside that maximum bound)?

I shall be very thankful for any guidance and suggestion. Thanks.