Does contractive operator $A$ have matrix norm smaller than 1 for every matrix norm? For every vector norm, $$||Ax||\leq ||x||$$. For spectral norm, or any induced norm, it is easy to see that contractive operator have norm smaller than or equal to one. But there are also other matrix norm that is not induced, right?
Does contractive operator have matrix norm smaller than or equal to 1 for every matrix norm?
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0An operator is contractive with respect to a given norm. Are you asking if, given that an operator is contractive with respect to one induced norm, then it is contractive with respect to all norms? Or perhaps at least all induced norms? – 2017-02-09
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0@Aweygan i think smaller than one is yes for induced norm. but there are other matrix norm that is not induced. – 2017-02-09
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0What do you mean by "i think smaller than one is yes for induced norm"? – 2017-02-09
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If $\|\cdot\|$ is a (matrix) norm then $c\|\cdot\|$ is a norm too for $c > 0$.
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0@Aweygan I think the OP is asking for matrix norms that are not induced. – 2017-02-09
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0Given the grammar, it is quite difficult to tell – 2017-02-09
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0since c can be as large as possible, why $c|||\cdot|||$ still smaller than 1? Let me denote $|||\cdot|||$ as matrix norm here. – 2017-02-09
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0@kww That is the point, it can't. – 2017-02-09
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0ok thanks. but this is true for every induced norm, right? – 2017-02-09
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0@kww A contracted operator is an operator whose **operator norm** is less or equal to $1$, so you are right. – 2017-02-09
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0ok, i havent realized that operator norm must be induced norm. – 2017-02-09