Norm of an operator in different spaces

Question

Given three Banach spaces $X$,$Y$ and $Z$ such that $Y \subset Z$ with dense and continuous embedding, we consider $T\in \mathcal{L}(X,Y)$. Is it true that $$ \|T \|_{\mathcal{L}(X,Y)} = \|T\|_{\mathcal{L}(X,Z)} $$ Thank you.

Answer 1

No. Let $X=Y=Z=\mathbb R^{2}$, give $X$ and $Y$ the Euclidean norm and $Z$ the norm $\|(x,y)\|=|x|+|y|$. Then the identity may gives a continuous dense embedding but the two norms for $T$ are $1$ and $\sqrt 2$.