Title says it all.
Is there any specific operator symbol for matrix multiplication?
Not just write down side by side but symbols like cross ($\times$).
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Juxtaposition is the standard notational convention (to "write side by side") without an intermediary operation symbol): for matrices $A, B$ on which matrix multiplication is defined, write $AB$. Some texts may use the "dot" $A\cdot B$, but juxtaposition is more typical.
See the entry in Wikipedia: Matrix Multiplication to disambiguate "standard" matrix multiplication from other matrix products: Hadamard product $A\circ B$, Frobenius product $A:B$, and Kronecker product $A\otimes B$.
You'll see the "dot" and "cross" typically used with vectors (which are, also, matrices) to distinguish between the operations of the dot product and the cross product.