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Q1 is already answered by Prof. Dirk Werner above. I simply list a number of equivalent conditions that seems to be related, but not the same as Grothendieck property.

The following are indeed equivalent:

1. every weak*-null sequence in $X^{\ast}$ is has a weakly Cauchy subsequence.
2. every bounded weak*-sequentially compact subset of $X^{\ast}$ is weakly precompact (every sequence has a weakly Cauchy subsequence).
3. for every bounded $T:X\to c_0$, the adjoint $T^{\ast}:\ell^1\to X^{\ast}$ is weakly precompact (i.e., $T^{\ast}$ maps bounded sets onto weakly precompact sets).
4. for every bounded $T:X\to Y$, where $Y$ is another Banach space with weak* sequentially compact dual ball, the adjoint $T^{\ast}$ is weakly precompact.
5. no weak*-null sequence in the unit ball of $X^{\ast}$ contains an $\ell^1$-subsequence.
6. there is no surjective bounded $T:X\to c_0$.

It is well known that $X$ is a Grothendieck space iff $X^{\ast}$ is weakly sequentially complete and (6)

Q1 is already answered by Prof. Dirk Werner above. I simply list a number of equivalent conditions that seems to be related, but not the same as the Grothendieck property.

The following are indeed equivalent:

1. every weak*-null sequence in $X^{\ast}$ has a weakly Cauchy subsequence.
2. every bounded weak*-sequentially compact subset of $X^{\ast}$ is weakly precompact (every sequence has a weakly Cauchy subsequence).
3. for every bounded $T:X\to c_0$, the adjoint $T^{\ast}:\ell^1\to X^{\ast}$ is weakly precompact (i.e., $T^{\ast}$ maps bounded sets onto weakly precompact sets).
4. for every bounded $T:X\to Y$, where $Y$ is another Banach space with weak* sequentially compact dual ball, the adjoint $T^{\ast}$ is weakly precompact.
5. no weak*-null sequence in the unit ball of $X^{\ast}$ contains an $\ell^1$-subsequence.
6. there is no surjective bounded $T:X\to c_0$.

It is well known that $X$ is a Grothendieck space iff $X^{\ast}$ is weakly sequentially complete and (6)

Q1 is already answered by Prof. Dirk Werner above. I simply list a number of equivalent conditions for convenience below.

The following are indeed equivalent:

1. every weak*-null sequence in $X^{\ast}$ is weakly Cauchy.
2. every bounded weak*-sequentially compact subset of $X^{\ast}$ is weakly precompact (every sequence has a weakly Cauchy subsequence).
3. for every bounded $T:X\to c_0$, the adjoint $T^{\ast}:\ell^1\to X^{\ast}$ is weakly precompact (i.e., $T^{\ast}$ maps bounded sets onto weakly precompact sets).
4. for every bounded $T:X\to Y$, where $Y$ is another Banach space with weak* sequentially compact dual ball, the adjoint $T^{\ast}$ is weakly precompact.
5. $X^{\ast}$ is weakly sequentially complete, and no weak*-null sequence in the unit ball of $X^{\ast}$ contains an $\ell^1$-subsequence.
6. $X^{\ast}$ is weakly sequentially complete, and there is no surjective bounded $T:X\to c_0$.

Q1 is already answered by Prof. Dirk Werner above. I simply list a number of equivalent conditions that seems to be related, but not the same as Grothendieck property.

The following are indeed equivalent:

1. every weak*-null sequence in $X^{\ast}$ is has a weakly Cauchy subsequence.
2. every bounded weak*-sequentially compact subset of $X^{\ast}$ is weakly precompact (every sequence has a weakly Cauchy subsequence).
3. for every bounded $T:X\to c_0$, the adjoint $T^{\ast}:\ell^1\to X^{\ast}$ is weakly precompact (i.e., $T^{\ast}$ maps bounded sets onto weakly precompact sets).
4. for every bounded $T:X\to Y$, where $Y$ is another Banach space with weak* sequentially compact dual ball, the adjoint $T^{\ast}$ is weakly precompact.
5. no weak*-null sequence in the unit ball of $X^{\ast}$ contains an $\ell^1$-subsequence.
6. there is no surjective bounded $T:X\to c_0$.

It is well known that $X$ is a Grothendieck space iff $X^{\ast}$ is weakly sequentially complete and (6)

Q1 is already answered by Prof. Dirk Werner above. I simply list a number of equivalent conditions for convenience below.

The following are indeed equivalent:

1. every weak*-null sequence in $X^{\ast}$ is weakly Cauchy.
2. every bounded weak*-sequentially compact subset of $X^{\ast}$ is weakly precompact (every sequence has a weakly Cauchy subsequence).
3. for every bounded $T:X\to c_0$, the adjoint $T^{\ast}:\ell^1\to X^{\ast}$ is weakly precompact (i.e., $T^{\ast}$ maps bounded sets onto weakly precompact sets).
4. for every bounded $T:X\to Y$, where $Y$ is another Banach space with weak* sequentially compact dual ball, the adjoint $T^{\ast}$ is weakly precompact.
5. $X^{\ast}$ is weakly sequentially complete, and no weak*-null sequence in the unit ball of $X^{\ast}$ contains an $\ell^1$-subsequence.
6. $X^{\ast}$ is weakly sequentially complete, and there is no surjective bounded $T:X\to c_0$.
7. $X^{\ast}$ is weakly sequentially complete, and $X$ contains no complemented copy of $c_0$.

Q1 is already answered by Prof. Dirk Werner above. I simply list a number of equivalent conditions for convenience below.

The following are indeed equivalent:

1. every weak*-null sequence in $X^{\ast}$ is weakly Cauchy.
2. every bounded weak*-sequentially compact subset of $X^{\ast}$ is weakly precompact (every sequence has a weakly Cauchy subsequence).
3. for every bounded $T:X\to c_0$, the adjoint $T^{\ast}:\ell^1\to X^{\ast}$ is weakly precompact (i.e., $T^{\ast}$ maps bounded sets onto weakly precompact sets).
4. for every bounded $T:X\to Y$, where $Y$ is another Banach space with weak* sequentially compact dual ball, the adjoint $T^{\ast}$ is weakly precompact.
5. $X^{\ast}$ is weakly sequentially complete, and no weak*-null sequence in the unit ball of $X^{\ast}$ contains an $\ell^1$-subsequence.
6. $X^{\ast}$ is weakly sequentially complete, and there is no surjective bounded $T:X\to c_0$.