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What can be said about the tensor product $T\otimes S$ of two Fredholm operatores $T:X_1\to Y_1$ and $S:X_2 \to Y_2$ where $X_1,X_2,Y_1, Y_2$ are Banach spaces and tensor product of operatores are considered on the projective tensor product of underling Banach spaces? Is the index of tensor product operator $T\otimes S$ bounded by the index of $T$ and index of $S$? What about if we replace the projective tensor product with other types of Banach space tensor product norm?

As a particular case what is a precise formula for the tensor product of two Laplacian operators on a Riemannian manifold? Namely assume that we have two Riemannian metrics $g_1,g_2$ on $M$ with corresponding Laplacian $\Delta_{g_1}, \Delta_{g_2}$. Is $\Delta_{g_1}\otimes \Delta_{g_2}$ a Laplacian of a metric on $M$?

What can be said about the tensor product $T\otimes S$ of two Fredholm operators $T:X_1\to Y_1$ and $S:X_2 \to Y_2$ where $X_1,X_2,Y_1, Y_2$ are Banach spaces and tensor product of operators are considered on the projective tensor product of underling Banach spaces? Is the index of tensor product operator $T\otimes S$ bounded by the index of $T$ and index of $S$? What about if we replace the projective tensor product with other types of Banach space tensor product norm?

As a particular case what is a precise formula for the tensor product of two Laplacian operators on a Riemannian manifold? Namely assume that we have two Riemannian metrics $g_1,g_2$ on $M$ with corresponding Laplacian $\Delta_{g_1}, \Delta_{g_2}$. Is $\Delta_{g_1}\otimes \Delta_{g_2}$ a Laplacian of a metric on $M$?

What can be said about the tensor product $T\otimes S$ of two Fredholm operatores $T:X_1\to Y_1$ and $S:X_2 \to Y_2$ where $X_1,X_2,Y_1, Y_2$ are Banach spaces and tensor product of operatores are considered on the projective tensor product of underling Banach spaces? Is the index of tensor product operator $T\otimes S$is bounded by the index of $T$ and index of $S$? What about if we replace the projective tensor product with other types of Banach space tensor product norm?

As a particular case what is a precise formula for the tensor product of two Laplacian operators on a Riemannian manifold? Namely assume that we have two Riemannian metrics $g_1,g_2$ on $M$ with corresponding Laplacian $\Delta_{g_1}, \Delta_{g_2}$. Is $\Delta_{g_1}\otimes \Delta_{g_2}$ a Laplacian of a metric on $M$?

What can be said about the tensor product $T\otimes S$ of two Fredholm operatores $T:X_1\to Y_1$ and $S:X_2 \to Y_2$ where $X_1,X_2,Y_1, Y_2$ are Banach spaces and tensor product of operatores are considered on the projective tensor product of underling Banach spaces? Is the index of tensor product operator $T\otimes S$ bounded by the index of $T$ and index of $S$? What about if we replace the projective tensor product with other types of Banach space tensor product norm?

As a particular case what is a precise formula for the tensor product of two Laplacian operators on a Riemannian manifold? Namely assume that we have two Riemannian metrics $g_1,g_2$ on $M$ with corresponding Laplacian $\Delta_{g_1}, \Delta_{g_2}$. Is $\Delta_{g_1}\otimes \Delta_{g_2}$ a Laplacian of a metric on $M$?

What can be said about the tensor product $T\otimes S$ of two Fredholm operatores $T:X_1\to Y_1$ and $S:X_2 \to Y_2$ where $X_1,X_2,Y_1, Y_2$ are Banach spaces and tensor product of operatores are considered on the projective tensor product of underling Banach spaces? Is the index of tensor product operator $T\otimes S$is bounded by the index of $T$ and index of $S$? What about if we replace the projective tensor product with other types of Banach space tensor product norm/

As a particular case what is a precise formula for the tensor product of two Laplacian operators on a Riemannian manifolds. Namely assume that we have two Riemannian metrics $g_1,g_2$ on $M$ with corresponding Laplacian $\Delta_{g_1}, \Delta_{g_2}$. Is $\Delta_{g_1}\otimes \Delta_{g_2}$ a Laplacian of a metric on $M$?

What can be said about the tensor product $T\otimes S$ of two Fredholm operatores $T:X_1\to Y_1$ and $S:X_2 \to Y_2$ where $X_1,X_2,Y_1, Y_2$ are Banach spaces and tensor product of operatores are considered on the projective tensor product of underling Banach spaces? Is the index of tensor product operator $T\otimes S$is bounded by the index of $T$ and index of $S$? What about if we replace the projective tensor product with other types of Banach space tensor product norm?

As a particular case what is a precise formula for the tensor product of two Laplacian operators on a Riemannian manifold? Namely assume that we have two Riemannian metrics $g_1,g_2$ on $M$ with corresponding Laplacian $\Delta_{g_1}, \Delta_{g_2}$. Is $\Delta_{g_1}\otimes \Delta_{g_2}$ a Laplacian of a metric on $M$?