Here are some general ramblings, which might be helpful. I have not read [2].

Let $f:M\rightarrow \mathbb R$ be a function. The Hessian of $f$ at $x$ is a bilinear map $\mathrm{hess} f: T_xM\times T_xM\rightarrow \mathbb R$. The Morse index can be defined as the maximal dimension of a subspace on which $\mathrm{hess} f$ is negative definite. Chosing a Riemannian metric (which can be subtle in the infinite dimensional contect), gives an isomorphism $T_xM\rightarrow T_x^*M$. One can use such an isomorphism to get an operator, also known as the hessian $\mathrm hess f:TxM\rightarrow T_xM$. The dimension of the negative eigenspace coincides with the Morse index defined before.

The index describes the behavior of $f$ in a neighborhood of the critical point. Indeed if the critical point is nondegenerate, the Morse lemma gives a normal form for the function. In this interpretation the index gives the number of directions in which $f$ is decreasing.

The eigenvectors of the Hessian (seen as an operator) point to the directions in which $f$ is decreasing. If you have a geometric understanding of $f$, then you can interpret this more geometrically.

For example if you take $M$ to be the space of paths between to given points in a Riemannian manifold $X$, and $f$ the energy functional, then the critical points correspond to geodesics between those points. The eigenvectors of negative eigenvalue are then vector fields along the geodesic, and have the following interpretation. If you exponentiate the curve along this vector field you end up with curves that are shorter than your geodesic (recall that geodesics are only locally length minimizing).

There is a whole story which ties these vector fields with Jacobi fields.

Here are some general ramblings, which might be helpful. I have not read [2].

Let $f:M\rightarrow \mathbb R$ be a function. The Hessian of $f$ at $x$ is a bilinear map $\mathrm{hess} f: T_xM\times T_xM\rightarrow \mathbb R$. The Morse index can be defined as the maximal dimension of a subspace on which $\mathrm{hess} f$ is negative definite. Chosing a Riemannian metric (which can be subtle in the infinite dimensional contect), gives an isomorphism $T_xM\rightarrow T_x^*M$. One can use such an isomorphism to get an operator, also known as the hessian $\mathrm hess f:T_xM\rightarrow T_xM$. The dimension of the negative eigenspace coincides with the Morse index defined before.

The index describes the behavior of $f$ in a neighborhood of the critical point. Indeed if the critical point is nondegenerate, the Morse lemma gives a normal form for the function. In this interpretation the index gives the number of directions in which $f$ is decreasing.

The eigenvectors of the Hessian (seen as an operator) point to the directions in which $f$ is decreasing. If you have a geometric understanding of $f$, then you can interpret this more geometrically.

For example if you take $M$ to be the space of paths between to given points in a Riemannian manifold $X$, and $f$ the energy functional, then the critical points correspond to geodesics between those points. The eigenvectors of negative eigenvalue are then vector fields along the geodesic, and have the following interpretation. If you exponentiate the curve along this vector field you end up with curves that are shorter than your geodesic (recall that geodesics are only locally length minimizing).

There is a whole story which ties these vector fields with Jacobi fields.