This paper investigates the use of multi-antennas in cognitive radio (CR) networks to achieve opportunistic spectrum sharing. In CR networks, secondary users (lower priority) can opportunistically use the spectrum allocated to primary users (higher priority) while minimizing interference. The paper formulates the design of a transmit spatial spectrum for a secondary user under both its own transmit-power constraint and interference-power constraints at primary receivers as a sequence of convex optimization problems. Convex optimization techniques are used to design algorithms for the optimal secondary transmit spatial spectrum, and suboptimal solutions are also presented for ease of implementation. The paper extends these algorithms to multi-channel transmission, allowing the secondary user to adapt transmit resources in space, time, and frequency for opportunistic spectrum sharing. Simulation results show that multi-antennas at the secondary transmitter can achieve substantial capacity gains even under stringent interference constraints. The paper also presents two suboptimal algorithms for MIMO channels: Direct-Channel SVD (D-SVD) and Projected-Channel SVD (P-SVD). These algorithms balance spatial multiplexing and interference avoidance. The D-SVD algorithm is optimal when interference constraints are inactive, while the P-SVD algorithm is optimal when interference constraints are active. The paper also shows that the P-SVD algorithm achieves a higher spatial multiplexing gain as transmit power increases. The results demonstrate that multi-antennas can significantly enhance the capacity of secondary users in CR networks while minimizing interference to primary users.This paper investigates the use of multi-antennas in cognitive radio (CR) networks to achieve opportunistic spectrum sharing. In CR networks, secondary users (lower priority) can opportunistically use the spectrum allocated to primary users (higher priority) while minimizing interference. The paper formulates the design of a transmit spatial spectrum for a secondary user under both its own transmit-power constraint and interference-power constraints at primary receivers as a sequence of convex optimization problems. Convex optimization techniques are used to design algorithms for the optimal secondary transmit spatial spectrum, and suboptimal solutions are also presented for ease of implementation. The paper extends these algorithms to multi-channel transmission, allowing the secondary user to adapt transmit resources in space, time, and frequency for opportunistic spectrum sharing. Simulation results show that multi-antennas at the secondary transmitter can achieve substantial capacity gains even under stringent interference constraints. The paper also presents two suboptimal algorithms for MIMO channels: Direct-Channel SVD (D-SVD) and Projected-Channel SVD (P-SVD). These algorithms balance spatial multiplexing and interference avoidance. The D-SVD algorithm is optimal when interference constraints are inactive, while the P-SVD algorithm is optimal when interference constraints are active. The paper also shows that the P-SVD algorithm achieves a higher spatial multiplexing gain as transmit power increases. The results demonstrate that multi-antennas can significantly enhance the capacity of secondary users in CR networks while minimizing interference to primary users.