Working with $SU(2)$, from Clebsch-Gordan coefficient tables I have for example: $$|j=3/2, m=3/2> \;= |1, 1> \otimes |1/2, 1/2> $$

How can I write $\hat{O}|j=3/2, m=3/2>$ in terms of $|1, 1 >\otimes| 1/2, 1/2>$ ?

My guess would be something like: $$\hat{O}|j=3/2, m=3/2> = (\mathbb{1} \otimes \hat{O} + \hat{O} \otimes \mathbb{1})|1, 1 >\otimes |1/2, 1/2> $$

Is this correct? Is this a valid question in the first place, or is such an operation undefined?

Working with $SU(2)$, from Clebsch-Gordan coefficient tables I have for example: $$|j=3/2, m=3/2\rangle \;= |1, 1\rangle \otimes |1/2, 1/2\rangle $$

How can I write $\hat{O}|j=3/2, m=3/2\rangle$ in terms of $|1, 1 \rangle \otimes| 1/2, 1/2\rangle$ ?

My guess would be something like: $$\hat{O}|j=3/2, m=3/2\rangle = (\mathbb{1} \otimes \hat{O} + \hat{O} \otimes \mathbb{1})|1, 1 \rangle\otimes |1/2, 1/2\rangle $$

Is this correct? Is this a valid question in the first place, or is such an operation undefined?

Working with SU(2), from Clebsch-Gordan coefficient tables I have for example: $$|j=3/2, m=3/2> \;= |1, 1 \otimes 1/2, 1/2> $$

How can I write $\hat{O}|j=3/2, m=3/2>$ in terms of $|1, 1 \otimes 1/2, 1/2>$ ?

My guess would be something like: $$\hat{O}|j=3/2, m=3/2> = (\mathbb{1} \otimes \hat{O} + \hat{O} \otimes \mathbb{1})|1, 1 \otimes 1/2, 1/2> $$

Is this correct  ? Is this a valid question in the first place, or is such an operation undefined  ?

Working with $SU(2)$, from Clebsch-Gordan coefficient tables I have for example: $$|j=3/2, m=3/2> \;= |1, 1> \otimes |1/2, 1/2> $$

How can I write $\hat{O}|j=3/2, m=3/2>$ in terms of $|1, 1 >\otimes| 1/2, 1/2>$ ?

My guess would be something like: $$\hat{O}|j=3/2, m=3/2> = (\mathbb{1} \otimes \hat{O} + \hat{O} \otimes \mathbb{1})|1, 1 >\otimes |1/2, 1/2> $$

Is this correct? Is this a valid question in the first place, or is such an operation undefined?