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improved latex formatting
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MTS
MTS
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supposeSuppose $\mathcal{H}$ is a separable Hilbert space over $\mathbb{C}$ (countable dimensions) with inner inner product $<,>$$\langle,\rangle$. Let $A$ be a bounded linear operator on $\mathcal{H}$, i.e, in B($\mathcal{H}$ $B(\mathcal{H}$). Suppose Suppose further that $A$ is not a multiple of the identity operator. Then is it true that there exist two elements of $\mathcal{H}$, call them $v_1$,$v_2$, of norm 1, such that $< v_1 , A v_1> \neq < v_2, A v_2>$.$\langle v_1 , A v_1 \rangle \neq \langle v_2, A v_2 \rangle$? This is true in finite dimensions (I think).

suppose $\mathcal{H}$ is a separable Hilbert space over $\mathbb{C}$ (countable dimensions) with inner product $<,>$. Let $A$ be a bounded linear operator on $\mathcal{H}$, i.e, in B($\mathcal{H}$). Suppose further that $A$ is not a multiple of the identity operator. Then is it true that there exist two elements of $\mathcal{H}$, call them $v_1$,$v_2$, of norm 1, such that $< v_1 , A v_1> \neq < v_2, A v_2>$. This is true in finite dimensions (I think).

Suppose $\mathcal{H}$ is a separable Hilbert space over $\mathbb{C}$ (countable dimensions) with inner product $\langle,\rangle$. Let $A$ be a bounded linear operator on $\mathcal{H}$, i.e, in $B(\mathcal{H}$). Suppose further that $A$ is not a multiple of the identity operator. Then is it true that there exist two elements of $\mathcal{H}$, call them $v_1$,$v_2$, of norm 1, such that $\langle v_1 , A v_1 \rangle \neq \langle v_2, A v_2 \rangle$? This is true in finite dimensions (I think).

added 18 characters in body
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magya_bloom
magya_bloom
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suppose $\mathcal{H}$ is a separable Hilbert space over $\mathbb{C}$ (countable dimensions) with inner product $<,>$. Let $A$ be a bounded linear operator on $\mathcal{H}$, i.e, in B($\mathcal{H}$). Suppose further that $A$ is not a multiple of the identity operator. Then is it true that there exist two elements of $\mathcal{H}$, call them $v_1$,$v_2$, of norm 1, such that $< v_1 , A v_1> \neq < v_2, A v_2>$. This is true in finite dimensions (I think).

suppose $\mathcal{H}$ is a separable Hilbert space (countable dimensions) with inner product $<,>$. Let $A$ be a bounded linear operator on $\mathcal{H}$, i.e, in B($\mathcal{H}$). Suppose further that $A$ is not a multiple of the identity operator. Then is it true that there exist two elements of $\mathcal{H}$, call them $v_1$,$v_2$, of norm 1, such that $< v_1 , A v_1> \neq < v_2, A v_2>$. This is true in finite dimensions (I think).

suppose $\mathcal{H}$ is a separable Hilbert space over $\mathbb{C}$ (countable dimensions) with inner product $<,>$. Let $A$ be a bounded linear operator on $\mathcal{H}$, i.e, in B($\mathcal{H}$). Suppose further that $A$ is not a multiple of the identity operator. Then is it true that there exist two elements of $\mathcal{H}$, call them $v_1$,$v_2$, of norm 1, such that $< v_1 , A v_1> \neq < v_2, A v_2>$. This is true in finite dimensions (I think).

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magya_bloom
magya_bloom
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non-Identity operator on a separable Hilbert space

suppose $\mathcal{H}$ is a separable Hilbert space (countable dimensions) with inner product $<,>$. Let $A$ be a bounded linear operator on $\mathcal{H}$, i.e, in B($\mathcal{H}$). Suppose further that $A$ is not a multiple of the identity operator. Then is it true that there exist two elements of $\mathcal{H}$, call them $v_1$,$v_2$, of norm 1, such that $< v_1 , A v_1> \neq < v_2, A v_2>$. This is true in finite dimensions (I think).