We consider a heat-structure interaction model where the structure is subject to viscoelastic (strong) damping, and where a (boundary) control acts at the interface between the two media in Neumann-type or Dirichlet-type conditions. For the boundary (interface) homogeneous case, the free dynamics generates a s.c. contraction semigroup which, moreover, is analytic on the energy space and exponentially stable here Lasiecka et al. (Pure Appl Anal, to appear). If A is the free dynamics operator, and B-N is the (unbounded) control operator in the case of Neumann control acting at the interface, it is shown that A(-1/2)B(N) is a bounded operator from the interface measured in the L-2-norm to the energy space. We reduce this problem to finding a characterization, or at least a suitable subspace, of the domain of the square root (-A)(1/2), i.e., D((-A)(1/2)), where A has highly coupled boundary conditions at the interface. To this end, here we prove that D((-A)(1/2)) equivalent to D((-A*)(1/2)) equivalent to V, with the space V explicitly characterized also in terms of the surviving boundary conditions. Thus, this physical model provides an example of a matrix-valued operator with highly coupled boundary conditions at the interface, where the so called Kato problem has a positive answer. (The well-known sufficient condition Lions in J Math Soc JAPAN 14(2):233-241, 1962, Theorem 6.1 is not applicable.) After this result, some critical consequences follow for the model under study. We explicitly note here three of them: (i) optimal (parabolic-type) boundary -> interior regularity with boundary control at the interface; (ii) optimal boundary control theory for the corresponding quadratic cost problem; (iii) min-max game theory problem with control/disturbance acting at the interface. On the other hand, if B-D is the (unbounded) control operator in the case of Dirichlet control acting at the interface, it is then shown here that A(-1)B(D) is a bounded operator from the interface measured this time in the H-1/2-norm to the energy space. Similar consequences follow.