Suppose, $ A $ is a unitary matrix in $ M_n(\mathbb{C}) $ given by $ (a_{i,j})_{1\le i,j\le n} $ which has the property that, for all the basis elements $ e_i $, $ Ae_i\ne |\lambda| e_j $ for all $i,j $ and $ |\lambda|=1 $. Then consider the matrix $ B=(a_{i,j}^4)_{1\le i,j\le n} $. Then is it true that $ ||B||< 1 $?

I could prove that, $ ||B||\le 1 $, but I want to know if its strictly less than 1 in this case.

Also is it possible to get such a matrix $ A $ for which $ ||B||=1 $ when $ A $ is a operator on a infinite dimensional Hilbert space with basis $ \{e_i\}_i $?

Suppose, $ A $ is a unitary matrix in $ M_n(\mathbb{C}) $ given by $ (a_{i,j})_{1\le i,j\le n} $ which has the property that, for all the basis elements $ e_i $, $ Ae_i\ne |\lambda| e_j $ for all $i,j $ and $ |\lambda|=1 $. Then consider the matrix $ B=(a_{i,j}^4)_{1\le i,j\le n} $. Then is it true that $ ||B||< 1 $?

I could prove that, $ ||B||\le 1 $, but I want to know if its strictly less than 1 in this case.

Also is it possible to get such a matrix $ A $ for which $ ||B||=1 $ and $ B $ actually attends its norm when $ A $ is a operator on a infinite dimensional Hilbert space with basis $ \{e_i\}_i $?

Edit: Will Sawin already posted a solution for the finite dimensional case. So I just edited the infinite dimensional case and added a condition I forgot to mention earlier.

Suppose, $ A $ is a unitary matrix in $ M_n(\mathbb{C}) $ given by $ (a_{i,j})_{1\le i,j\le n} $ which has the property that, for all the basis elements $ e_i $, $ Ae_i\ne e_j $ for all $i,j $. Then consider the matrix $ B=(a_{i,j}^4)_{1\le i,j\le n} $. Then is it true that $ ||B||< 1 $?

I could prove that, $ ||B||\le 1 $, but I want to know if its strictly less than 1 in this case.

Also is it possible to get such a matrix $ A $ for which $ ||B||=1 $ when $ A $ is a operator on a infinite dimensional Hilbert space with basis $ \{e_i\}_i $?

Suppose, $ A $ is a unitary matrix in $ M_n(\mathbb{C}) $ given by $ (a_{i,j})_{1\le i,j\le n} $ which has the property that, for all the basis elements $ e_i $, $ Ae_i\ne |\lambda| e_j $ for all $i,j $ and $ |\lambda|=1 $. Then consider the matrix $ B=(a_{i,j}^4)_{1\le i,j\le n} $. Then is it true that $ ||B||< 1 $?

I could prove that, $ ||B||\le 1 $, but I want to know if its strictly less than 1 in this case.

Also is it possible to get such a matrix $ A $ for which $ ||B||=1 $ when $ A $ is a operator on a infinite dimensional Hilbert space with basis $ \{e_i\}_i $?